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A DYNAMIC ECONOMETRIC MODEL OF THE U.S. SHRIMP MARKET By JOHN DALE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1988 ACKNOWLEDGEMENTS wish to express sincere appreciation to Dr. Scott Shonkwiler diligent, helpful, cooperative guidance maj or professor during the development of this dissertation, two presentation papers, extra Grant proposal. wish express appreciation to Drs. Maddala, C. Cato , R. L. Kilmer, and T . G. Taylor dedicating time serve on my advisory committee and for their helpful comments during preparation of the dissertation. would like acknowledge assistance received from Drs. John Vondruska and John Poffenberger the National Marine Fisheries Service. These gentlemen provided most of the data used in the study. would especially like thank Roberts of Louisiana State University allowing use, misspecification test demonstrated herein, leadership. simultaneous Roberts also equation provided model data developed used under to estimate their model. Special thanks to Dr. "Chuck" Adams very tolerantly allowed me also also office provided goes space a great to Dr. Jim access deal Cato to his library of friendship. and the Florida My very Sea Gran and his sincere t Program computer. appreciation and Dr. Max Langham, John Gordon, Larry Libby of the Food and Resource Economic s Department financial suDDOrt that made Ph.D. Finally would like thank mother, Mrs. Luana Lea; mother father inlaw Mrs. Shattuck; children, Christopher, Georgia, and Margaret; rest extended family for their moral support during this unusual adventure. greatest appreciation goes wife, Marion, unfaltering love support through these four years of intense study. thank for the many sacrifices she has made to allow me pursue my goals. could not have done it without Marion. TABLE OF CONTENTS ACKNOWLEDGEMENTS . . ..... .... .... LIST OF TABLES .. a a .. a a a a a a a a * CHAPTERS INTRODUCTION.... ....... . . .. . . 1 An Introduction Problem Research to the Shrimp Market Statement. . t . . .. 1 ,. . ... 7 10 ectives Research Approach............. The Application of the Results of This Study Scientific Policy view of Merit... Applications the Remaini * S t C. C nf Li Chapters * S S C C S C * t e C S C * . . . . . . ..Ug l .. .. .. REVIEW METHODOLOGICAL CONSIDERATIONS. .......e............. ...... Data Constraints.. . .. .. . . .... 25 Some Policy The Adequacy Implications of of the Availabl Inadequate Data Data as Proxies . .... 25 for the Desired Data... .... 27 Empirical Available Evidence Proxies.. Relating to the Inadequacy Implications of Inadequate Data for Previous Studies Research Re Theoretical source Constr Constraints aints..... Policy Implications Implications of Misspecification for the Pr esent Effort MISSPECIFICATION TEST . . ..... .... .... Hausman Test. . a. a. . t. a a o I t 4 1 Pre ... 29 .......... ... 34 ...... ..... ....... 35 Page AB STRA CT .................................. ..... .... .. OF LITERATURE .............. ..... ..... .... tlll(~ I Summary. ...... ..... . .. .... . . . VECTOR AUTOREGRESSIVE MODELS................................ Vector Autoregressive Theory Estimation Procedures.......... Determining the Order of the M Hosking's Test for White Noise Estimating Relationships Among Va Decomposition of the Parameter Frequency Dimension Analysis.. Time Dimension Analysis....... ESTIMATING odel * a a . r .  I * t a a ft 4 4 9 ft a a a a a a a *. a a. a a .. . t *. a p. .t iaDies Matrix THE VAR MODEL... ... ........ ................... action acting of Variables the Order of he VAR Model VAR Model... Testing the VAR Model.... Analyzing the VAR Model.. Frequency Domain Analysis Time Domain Analysis..... RESPECIFYING THE S IMULTANEOU S EQUATION MODEL........... ...... Respecified Model . . . .. . .. . . 94 cussion of Analytical Results......................... 101 SEM Parameter Estimates .. ....... .... .. .... 101 Prediction Experiments .. . . .... .. . 107 Exogeneity of Imports . . .. .. . . 109 lity of the VAR Model .................. .... ... ....11 i VIII SUMMARY AND CONCLUSIONS ..... ... .. ....... ...... ........ Econometric Misspec Reduced Frequen lytical icy Imp Techniques .... .. .. .. .. .. .. ..... 115 ification Test.......... SForms. .. .. .. .. ..... cy and Time Domain Analy Re lic suits... ations.. * Vector Autoregressive Model Simultaneous Equation Model The Benefits of This Approach Concluding Remarks............ Pol: Pol: to S] e .. ... . 11 . .. ... ... t.. 11 ses. ..... .....11 . ...... .. ... .... 11 S............ 11 icy Implications........ 11 icy Implications........ 12 EM Model Specification.. 12 . ... .... ...... ... .. 12 APPENDICES SAS PROGRAM TO THOMPSON ET AL. CONDUCT THE MISSPECIFICATION TEST MODEL. .. .. ....... .. ... ... ON THE a .a .a. ..a PROGRAM TO ESTIMATE THE VAR MODEL AND ANALYZE ITS PARAMETER  d SAS PROC MATRIX TO SELECT THE ORDER OF THE VAR model......... AUTOCORRELATIONS AMONG VARIABLES IN THE VAR VECTOR AND ASSOCIATED PREDICTION ZVALUES.. EXPERIMENT RESULTS DATA USED IN THE RESPECIFIED OF THE SHRIMP MARKET... REFERE NC S .. ... .. ... .. ... .. ... .. .. .. .. .. .. .. .. BIOGRAPHICAL SKETCH.. .. ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. ............. LIST OF TABLES Table Page Comparing Restricted Reduced Form with Unrestricted Reduced Form Pa Parameter rameter Estimates Estimates...... Tests to Select the Order of the VAR Model.............. VAR Model Restricted Parameter Estimates tValues... Frequency Relationships of the VAR Model. .... .......... Autocorrelations 3SLS Parameter Estimates and Associated tValues........ Long Run Multipliers of the Complete Respecified SEM.... Prediction eriment 3SLS Parameter Estimates from the Four Equation SEM..... Long Run Multipliers of the Four Equation SEM........... Elasticity Flexibility estimates the Thompson et al. Respe Model cified Compared SEM. ... with Similar Estimates from Figures Page Selected U.S Shrimp Market in the VAR Model ... .. ..... .... . Results............... Data .... ..... .. .. .. . Abstract of Dissertation of the University of Florida Presented to the Graduate in Partial Fulfillment School of the Requirements for the Degree of Doctor of Philosophy A DYNAMIC THE U.S ECONOMETRIC SHRIMP MODEL MARKET John Dale L( April 1988 Chairman: Department: Scott Food Shonkwiler d Resource Economics Recently published statements relating fragility traditional simultaneous equation econometric models cast doubt on the specification of existing models therefore on the policies implied the models. societal I Concern welfare over suggests the impact that inappropriate p specification policiess can have models that might likely be relied upon for policy development should be tested for misspecification. Since the goal of such action is the development appropriate policy the objective a specific test is either to add creditability the existing model use the information obtained from test an effort to develop a more appropriately specified model. simultaneous econometric model (SEM) testing respecification process demonstrated here includes test with three stage least squares techniques. misspecification test modification test suggested Hausman. Here difference between unrestricted restricted reduced forms implied the structural model provide the data for the test. These data also provide clues sources of misspecification in the SEM being tested. A vector autoregressive model (VAR) of the U.S. shrimp market estimated part effort obtain information respecification of the SEM. Time frequency domain analyses model provide information useful policy purposes; however, relatively little of the information is useful in respecifying the SEM. policy implications model exis ting respecified SEMs are discussed. Also, implications existing data constraints on attempts to understand the U.S. shrimp market discussed. Although the existing SEM was found to be misspecified, its policy implications appear accurate. Analysis model indicated that shrimp imports lead domestic prices short term domestic prices over three year period. This implies significant amount inertia current situation of increasing imports decreasing domestic ices that has important implications for domestic foreign policy makers investors. are CHAPTER I INTRODUCTION An Introduction to the U.S . Shrimp Market The shrimp fishery is the most valuable of U.S fisheries terms exvesse revenues and in terms of the value of processed products. 1985 U.S. shrimp fishermen received approximately million dollars for their products. purposes of comparison, next most valuable U.S. fishery 1985 was Pacific salmon fishery with landings worth approximately million dollars, followed American lobster fishery with landings worth approximately 115 million dollars. Comparing shrimp industry with other food industries difficult due to the impacts it has over several states; however, it IS interesting note that Florida citrus growers received over million dollars for their 1983 orange crop (Mulkey et al.). The value U.S. processed shrimp products 1985 was approximately billion dollars represent approximately percent total value of U processed fishery products (USDCa, USDCb). addition importance national economy, domestic shrimp industry a major component of the economy in coastal communities of the Gulf and South Atlantic regions of the U.S. Griffin and Jones estimated the economic impact of the commercial shrimp catch on the Texas economy. According to their analysis, $1.00 output one supporting directed industries. toward substantial the construction portion maintenance this support of the fishing fleet. The Gulf South Atlantic fleet more than doubled in size from about 6,600 boats and vessels in 1950 to about 14.000 in 1983 (Vondrus ka). This increase in fleet size has occurred while total landings from area have grown less than percent: Southern landings in 1960 were pounds was 149 million in 1985 143 million pounds, (heads o pounds. 142 million >ff basis). T1 The average pounds average for the f 1983 , and 183 million for the 195083 iveyear period period ending with 1985 was 165 million pounds (USDCa). A major reason the fleet been able to expand in the face of such meager expansion in landings has been the increasing value of the catch. In real terms , the value of the 1983 catch , a year of reduced landings , was 3 times the value of the 1950 catch (Vondruska) Shrimp the Pacific are also regions. landed In fact, in northern most ports of the reported the New England increases in total landings during twenty year period between 1960 1980 were due to increases in these areas (Hu) However U.S. landings outside the Gulf South Atlantic region have remained fairly unimportant terms value and volume of landings. Over the five year period from 1980 to 1985 , landings from the northern area accounted for less than percent total landings (USDCa). Additionally, the shrimp landed in northern ports are a different species from those landed warm water shrimp landed southern ports can analyzed separately from that dealing with the cold water shrimp of the north. Perhaps, most important feature of the U.S shrimp market reliance on imported shrimp. Since early 1960s imports have been imports maj or of shrimp source have shrimp increased consumed dramatically, U.S. reaching Since a peak 1981 in 1983 when they represented over percent of U.S. supplies (Prochaska Keithly). Imports have continued increase every year since 1983 however the proportion of total U.S. shrimp supplies accounted for by imported shrimp declined increased landings approxi mately percent in 1986. Figure displays the landings imports data graphically. Also displayed are data on the real average monthly wholesale price restaurants 2630 other count eating shrimp places. data Because expenditures U.S. Japan major world markets shrimp, imports into U.S. are con sidered to be influenced the relationship between the value of the dollar and Japanese (Prochaska Keithly). worldwide basis, stocks naturally occurring shrimp thought to be fully exploited (Rackowe). During past decade, 1970 world have shrimp remained catch relatively increased stable over since percent; 1977 however, (Rackowe). landings Environmental conditions are considered the primary cause fluctuations in land ings. The annual shrimp crop spends a critical stage of its develop ment estuarine waters where rainfall water temperature are are can ten million pound units 3 2 1 3.7 2.2  1977 1982 1986 1la dollars __ : 1977 1982 1986 Figure elected Monthly Average Real Monthly Shrimp Shrimp Market Imports Wholesale Expenditures Data into Price source the U.S. 2630 in Eating see Appendix b) Monthly Count Places.; Shrimp.; d) Month Lly billion dollar units 4.6 3.6 2.6 2.0 0.2 1977 1982 1986 1lc ten million pound, headsoff units ,  aL 1977 1982 1986 1ld Figure .lcontinued While U.S. landings have remained fairly stable since the 1950s, U.S. consumption of shrimp has grown significantly. annual average rate of increase between 1950 1983 is 3.9 percent. In 1983 , U.S. consumption was times 1950 evel (Vondruska). capital consumption, however increased only about percent year over the twenty year period, 196685. The majority of this consumption, proximately percent occurred commercial eating establish ments nmentl , generally t al or company .ermed "institutions cafeterias. " such The fact that as restaurants data or gover on transactions the institutional major setting have impact not been economic collected analysis on a systematic shrimp basis market. Because essential data demand analysis are not available, attempts to estimate the retail demand for shrimp in the U.S. have produced misleading results. essential information for estimating the retail demand consumer product are data on the price paid consumer quantities institutional purchased. Given setting the preponderance relevant data of transactions calculating in the retail demand for shrimp the related elasticities of demand are data from transactions institutions. Unfortunately, prices shrimp sold through retail outlets such as supermarkets seafood markets are the only data available which relate directly transactions with final consumers. These data series are not complete because their collection discontinued early 1980s. available data was movements of shrimp out of wholesale warehouses. An alternative term for this movement is "disappearances from cold stora Another implication of the large proportion of institutional sales that shrimp demand is heavily dependent on the public s demand food away from home. Thus, disposable income is expected to be a key determinant of shrimp demand (Rackowe). Problem Statement With world shrimp catch levels near their expected maximum with reasonable expectations that the demand for shrimp will continue strengthen, a scenario that includes rising shr imp prices increased congestion U.S. shrimping grounds not unreasonable. However, during early 1980s, technology of shrimp production substantially transformed perfection commercialscale shrimp mariculture that ostensibly independent critical inputs from the natural environment. Accordingly, limits production shrimp have been expanded from that a natural production system to that of controlled production systems similar those used to produce swine or poultry. Shrimp are now being reared from breeding stock maturity highlands Colorado (Brannon, 1986). use this technology under more favorable biolo ical economic conditions existing outside implies that limits production are those imposed public domestic shrimp policy fishery market. are immediately ramifications obvious. was the marginal value (price) received foreign producers is just equal their marginal cost of production and distribution. To the extent that maricultured shrimp are less costly to produce than domestically captured shrimp in the absence of sufficient shifts in U.S demand for shrimp, these increased supplies will have a depressing effect the prices U.S. producers receive for shrimp. Because U.S. shrimp fishery considered fully developed possibly overcapitalized, purely competitive industry, decrease shrimp prices can expected force some U.S. fishermen out of business. Policy makers be called upon to devise programs that will assist U.S. citizens adjust these changing conditions. These policies should be based on accurate predictions the likely change s that will occur in the U.S. shrimp market due to the changed volume imported shrimp. Recent studies of the U.S. shrimp sector have contributed substan tially to the understanding of the industry. However, the development complete mathematical models of the industry useful for predicting price changes been severely hampered absence required data capture series many limitations of the complexities previous modeling of the market. techniques absence of ideal data series have resulted erroneous conclusions about such policy variables price elasticity demand income elasticity of demand. purpose of this paper to focus attention on this deficiency its implications. disequilibrium in product or factor markets, and pricing dynamics. suggested method treating such complexities vector autore gressive specification (Sims). Thus, a second purpose this paper investigate utility of the vector autoregressive specification in understanding the U.S. shrimp market. difficulty of modelling complex behavioral relationships economic process such a market have resulted misspecification of economic models of the U.S. shrimp market. Indeed economic model misspecification appears "more likely rule rather has been vary than the exception" cited between as a major studies (Judge reason model (Prochaska al. , 854) results and Keithly, Misspecification policy implications Adams, These considerations have important implications since policies based misspecified models be inappropriate. concern impacts which policy can have societal welfare suggests that responsible action take test specification models that might likely relied policy purposes. goal such action the development appropriate policy. objective creditability existing a specific test model either utilize information contained in the model and the information obtained from a test of its specification an effort to develop a more accurately specified model for policy making purposes. Technological development in the production of shrimp has allowed dislocation highly competitive U.S. production subsector substantial changes in the U.S. marketing subsector. Because policies developed address this situation will quite likely based existing econometric models the U.S. shrimp market an appropriate subject demonstration econometric model testing respecification procedure briefly described above. Research Objectives The major goal of this research to develop use in policy making, a more complete understanding underlying behavioral relationships demonstrate U.S. econometric shrimp market. techniques, secondary viz. , goal vector autoreg ressive model simultaneous equation model cification test, that prove useful additions analytical process understanding markets in general. The primary objective research to develop a dynamic, econometric model U.S. shrimp market which can judged being properly specified specification criteria to be discussed study. accomplishing this objective, identified deficiencies of previous models will be addressed. vector autoreg ressive specification market will investigated usefulness in understanding relations between variables of inter , for fulness usefulness suggesting predicting alterations instrument, structural econometric model which treat complex behavioral relationships addressed lack of ideal data will be addressed recognizing that existing data appropriate estimating retail level relationships. Instead, data will be used to estimate relation ships at the wholesale market level which generated the data. primary specification objective an existing will be accomplished econometric model first testing the market. Next vector autoregressive model of the market will be specified, estimated, analyzed. The information gained from these analyses will be used to specify a dynamic structural econometric model. The model will then tested for misspecification, forecast performance will compared with vector autore gressive model. Finally, policy implications of the analyses The specific objectives will of this be discussed. study are test systemwide econometric model of the U.S. specification shrimp existing market; to construct a vector autoregressive model the U.S. shrimp market; to determine point estimates for the temporal relation ships between landings, and wholesale import shrimp shrimp prices, U.S. quantities, shrimp construct confidence intervals to specify test for these estimates. the specification a traditional econometric model, using the results from objectives discuss policy implications models other information developed. Research Approach Testing specification an existing structural model of the U.S. shrimp market should provide an initial base of information useful further analysis. A specification test based on Hausman' s 1978 work developed applied. The difference between estimated parameters of the restricted unrestricted reduced forms implied structural ference model between provide two the basic reduced forms data also test. provides dif information possible sources of misspecification. Similarly, the specification the shrimp market vector autoregres sive form contributes additional information that is useful in respecifying the traditional simultaneous econometric model (SEM). The specification of vector autoregressive (VAR) model will based policy knowledge makers, U.S. economic shrimp theory. industry, likely degree concerns lags used spec ifying the VAR are selected a process of increasing the number of lags employed until tests of the hypothesis that parameters the additional set of lagged variables in the system are equal zero cannot be rejected (Nickelsburg). This test is executed in the context of least squares estimation procedures s since, in this case where each equation system contains identical regressors , least squares estimation is equivalent to maximum likelihood estimation (conditional number of lags specified will be the same for each equation, protesting will be limited to systemwide specification. Once order been selected, the model will reestimated a restricted form reflecting theoretical judgement relating the exogeneity of the variables in the endogenous system. The model will then be subjected a diagnostic test to determine distributional characteristics error terms system. vector variables associated structure have been selected appropriately, error terms should random processes. The degree to which they approach this definition will be tested using Hosking's portmanteau statistic (Hosking). hypothesis serially correlated residuals cannot rej ected addition variables to the model or some transformation of existing variables be indicated. Once vector autoregressive system been specified, relationships among variables of interest will be obtained decompos parameter matrix system, using analyses in the frequency time domains. Confidence intervals for these estimated relation ships will be constructed numerical differentiation. implications vector autoregressive model, along with information gained from testing specification existing structural model will be used in specifying a dynamic structural model (herein called "respecified" model distinguish from previously existing model of Thompson al. , which referred structural model. Finally, structural vector autoregressive models will be compared and the implications for policy discussed. The Application of the Results of This Studv Scientific Merit scientific merit of this study arises from the demonstration a model the econometric extent specification model use of this ied econometric models test development procedure and more can procedure results appropriate easily of other more police adopted analysts. as part To the specif general will benefit. Additional scientific merit practical utility derives from the demonstration, as a part of the testing respecifi cation process, a vector autoregressive specification phenomenon under study. Policy Applications Potential users investors, of the results foreign of this governments, study domestic include state domestic federal fishery regulatory agencies domestic international trade regulatory agency ies. The policy issues that can be addressed using the results of this study include the advisability a program to assist the U.S. shrimp production sector adjust projected, continued decrease real (adjusted inflation) shrimp prices possible, con comitant reduction in operating shrimping vess The results of the vector autoregressive analysis should be helpful in understanding which appropriately y, society in foreign Such understanding should helpful development appropriate policy. For example, given the impact of increased shrimp supplies shrimp prices indication delay industry' s response price changes, domestic foreign investors need to adjust their estimates of expected prices used in analyzing investments in additional productive capacity. The domestic marketing sector should find these results helpful in forming expectations of the likely increase in shrimp supply in planning investments to market the increase. Preview of the Remaining: Chapters The following chapter reviews the literature relating to existing studies using the shrimp existing market. data The third estimate chapter retail addresses level the problem relationships discusses other methodological considerations. fourth fifth chapters discuss the methodology used in applying the misspecification test in specifying vector autoregressive model in analyze dynamic relationships among the variables VAR. The sixth chapter discusses estimation results. seventh chapter discusses the reestimation of the structural economet ric model existing and and its estimated the respecified parameters. SEM results The final discusses chapter compares the policy implica tions models discusses benefits demonstrated approach development appropriately specified simultaneous equation models. CHAPTER REVIEW OF LITERATURE A highly valued industry allotted a corresponding degree interest collection from public of data officials. that makes This econometric interest analysis often results possible. Accord ingly, the U.S. shrimp industry has been the subject a substantial amount analysis. Naturally, one would expect progression methodologies employed over years to reflect developments economic theory, in statistics, in computing capability that have generally expect proceeded that their analysts application conducting in applied new studies economics. have One would attempted employ most advanced techniques at their disposal in their attempt common body of knowledge. This process tended increase this stock of knowledge, although perhaps not uniformly over time. Twenty years ago, most econometric models developed to analyze demand shrimp were singleequation functions. Nash catalogued "best specimen" demand functions selected, from those available 1968 a panel marine economists. demand functions selected shrimp were generally linear, single equation models estimated with least squares techniques. Three seven selected equations related the wholesale market level two related the P.xvpAsl market lpvsl nnd twn to the rstsil 1 eVP 1 The nrie L. I from .78 to +.38 with four out of the six ranging from .29 .46. retail price elasticity estimates were +.38 .28. Thus, none of the selected models estimated shrimp demand to be price elastic at the selected market level. The income elasticity estimates were in the elastic range .e. , greater than one ranged from .329 (Nash and Bell, 2226). next phase of model development appears to have been that multiequation models. example, Timmer; Gillespie Doll Batie; Hopkins et al. Having been published in the American Journal of Agricultural Economics, the study Doll perhaps, most widely known this group. discussing fiveequation simultaneous equation model based on annual data, Doll first deals with the maj or data problem confronting variables all researchers theoretically in this a: specified rea: the lack model. Doll briefly describes what variables an ideal model would contain and then states: realities "However, when available ideal data, some model was compromises confronted became with necessary" 432). One of those compromises involves imports as a source of shrimp the U.S. market. Although Doll recognizes that foreign suppliers have a choice of marketing areas that some U.S. importers on forward contracts, Doll forced treat imports an exogenous variable, lack of data. For a similar reason, Doll enters domestic landings shrimp into model exogenous variable. Doll two al.; compromises lend credence to the statement Sims that analysts often define a variable as exogenous because serious effort to model variable would take the researcher too far afield from his main area interest. Another compromise recursive nature model. Doll does not attempt to justify this specification. Continuing with the results reported by Doll, he shows that, as of 1968 domestic landings were single largest source price variation the U.S. shrimp industry. Increases in U.S. disposable income kept a constant upward pressure on shrimp prices that would have increased prices cents pound year. However increased import levels resulted an average annual decrease in prices of .05 cents per pound that partially offset the price increases due to income growth. Doll reports, with some caveats confidence estimates income elasticity of shrimp retail level to be 1.12 estimated own price demand elasticity at the retail level .63. Doll notes that although consumption shrimp individuals would be expected to be more responsive to retail price, such a result might be obscured in analyses, such as his, which have data majority market transactions involving final consumers. The institutional demand for shrimp comprises an important part the final market, cost of preparation other services would food. to changes important Consumption in price under alone determinant these the price conditions . (Doll, would not the prepared be responsive p. 435) not interest have attempted to model those variables that previously been through recognized some as being compromise. endogenous Also , the were dynamic treated relationships as exogenous between variables are given some recognition through use of lagged vari ables. Timmer incorporates import supply function while still regarding a domestic Gulf domestic supply of Mexico landings function water as exogenous. Batie 1958 temperatures successfully period to capture using biological estimated index fluctua tions in shrimp populations. Both Hopkins et al. Thompson et al. model domestic landings and imports. Hopkins develop annual model of domestic supply function of wholesale prices of shrimp lagged one year, Gulf South Atlantic landings shrimp lagged one year, index vessel operating costs lagged one year average monthly Mississippi River discharge for February through April. Import supply is modeled as a function of U.S. exvessel price lagged one year imports lagged year. Price dependent demand the wholesale level is modeled as a function Although beginning Hopkins stocks, imports, expected landings inverse relationship shrimp. between quantity supplied and price , they found a positive relationship between imports unexpected U.S. positive wholesale sign price. reflect authors a supply explained response that general market conditions. contradicts harvester The authors rs' complaint also that commented imports that have the positive a depressant sign effect one Beginning stocks of shrimp were found to have the largest direct impact (negative expected) wholesale prices. , beginning stocks are a function of imports, most of which enter the U.S during last quarter year (Hopkins al. , also valuable equation note that primarily researchers a time trend found with that lagged import imports supply playing dominant role in the determination current imports. As for elas ticities, Hopkins report income elasticity of 1 price elasticity .27. Thompson et al. model domestic landings as a function of fishing effort, average average precipitation atmospheric temperature in coastal in coastal Louisiana Louisiana lagged lagged two months, two months, quarterly dummy variables. Fishing effort a function current price exvessel esel price fuel, shrimp, seasonal current dummy quantity landed, variables. import quantity variable is modeled as a function the wholesale price shrimp lagged two months, rate of exchange between U.S. dollars the Japanese lagged two months, and seasonal dummy variables. Thompson suggest that studies the U.S. shrimp market using data collected prior to recent structural changes in the market be dated and, therefore, estimate a simultaneous system model using monthly data from September 1974 through December 1983. authors recognize difficulty estimating retail demand shrimp when the existing data relate to the wholesale level. It is difficult own In discussing their demand equation, they apparently consider retail demand function because they mention that "economic theory suggests that prices substitute complementary products included equation Equation their consumption equation. They also mention the results of adding the price index processed meat explanatory variable equation. reported version their model these variables were included because authors found variables to be insignificant trial runs. attempted inclusion of the processed meat variable could interpreted imply they consider equation representative derived wholesalelevel demand, processed meat could con sidered input a shrimp product. This consideration seems unlikely , impling that they consider the equation to be representative of retail demand. appears that Thompson . have attempted estimate retail demand using wholesalelevel data Thompson estimate price elasticity their demand function elasticity to be inelastic of demand that .11. contradicts They obtain that most an estimate other of income research. Thompson et al. estimate is in the inelastic range , viz .42 which indicative an inferior good. The authors explain this discrepancy noting that the elasticities from previous research were calculated from models estimated with data from the late 1950s to early 1970s: period when shrimp consumption was increasing. A portion dur inp the of the II ri od correlation I tl rn II  between nrPvi nijjq consumption stiidi sP dca1t wi th income pnn bP i  period would elasticity annual data of demand more demand. from 1960 shrimp closely approximate A recent through 1980 of 0.73. T study by found an actual Hu (1 income estimated L983) income using elasticity elasticity less than one can probably be attributed to the inclusion several years determinant of data after expenditures the early at retail 1970' eating With income establishments 7080 site percent use : shrimp income I consumption iv overstate occurring the size these elasticity (Thompson et Thompson estimate actually elasticity consumption Thus given the comparisc a change in between in expenditures estimates retail is difficult. eating places. expenditures variable alternative is used definition a proxy income for the variable authors. is possible. Since However, no quantity data accompany the expenditures data, it is impossible to determine changes expenditures are result changes in quantity sold, changes in p If the major proxy rice, chang for income or some ;e had i combination ,een is plausible. of changes in quantity, If the major in quantity use of change expenditure had been price. s as a in price, use of expenditures as a proxy for price becomes plausible. Under alternative interpretation that retail expenditures eating aces be considered a proxy output price, the Thompson et al. model might be expected to predict price elasticity rather than income elasticity. In this case, Thompson et al. estimate is consistent with the majority of estimates of price elasticity. most recent innovation in shrimp market modeling efforts utilization causality measures Adams assist n specifying S, p. be included test in the hypotheses price determination dynamic properties model and regarding i between of price d then use these lead/lag interfacin [eterminatii stochastic structures g market on and the characteristics and the direction levels. Finally, structural attributes market, suggested theory, are incorporated into econometric model describing price at each market level. Specifically, causality tests are used specify a model based on monthly data a recursive form while a simultaneous form dictated for a model based on quarterly data. Theoretical considera tions determined what nonprice variables, such income, quantities, and consumer price index, would enter each of the structural equations. These nonprice variables enter the equations an unlagged or current form. The causality testing procedures, their related impulse response functions, variables and and certain their diagnostic specific checks degree dictate lag e which nter price structural equations. Adams' s results relating income elasticity support those Thompson et al.: Real disposable income was found not a significant determinant of monthly quarterly prices either size ass. This finding reflects the fact that real disposable income over changes e time very period little on a monthly the analysis. or quarterly Previous basis monthly quarterly analyses corroborate insignificance income while studies using annual data typically find income significant. (Adams, 165) The price flexibilities of demand derived from final form coeffi cient' n a re ouite inelastic less than . i .e . for both size classes . 2 4m I general, most variables anticipated relationship with dependent variable of their related , price dependent demand equations. However, termed: there was all other one exce imports. ption This involving variable one category a positive imports relationship with the dependent variable of both size classes of shrimp under study. Adams found that beginning inventories a larger effect both classes landings. shrimp This prices effect than does is in accord class with of imports the findings or domestic of Hopkins et al. Also in accord with Hopkins et al ., Adams found that total imports shrimp ar quarterly e positively models, related not to price. a significant Income, determinax in Adams's It of price monthly dependent demand. CHAPTER METHODOLOGICAL CONSIDERATIONS Economists strive develop their models on strong theoretical foundations; resource however, constraints, limitations availability economic of data theory, can often research lead model misspecifications chapter will discuss which reduce the implications utility of these three model. constraints This on the present past efforts to better understand the U.S. shrimp market. Data Constraints maj or data constraint, mentioned above lack information on quantities and prices of shrimp sold to final consumers through institutional markets. Although approximately percent U.S. shrimp consumption occurs these commercial eating places price quantity data these transactions are systematically collected. a result, researchers are forced use available data from sales retail outlets, which accounts only percent shrimp consumption, represent the primary market data (Adams, Thompson et al., 12). Alternatively, analysts chose estimate retail level relationships using prices quantities from the wholesale level. Some Policy Implications of Inadequate Data Neither these two approaches appropriate data used are not adeauate nroxi es for the desired data . If the data . say data, then relationships found more accurately describe wholesale level rather than the retail level. Using wholesale level relationships development lieu of policy can lead desired retail to inappropriate relationships policies. an example, consider a situation in which an industry characterized constant returns scale fixed proportions. Layard Walters' s (p. 260276) illustration Marshall' s rules relating to the influences governing the price elasticity of demand a factor proportion of production production helpful function this perfect context. competition Given in the fixed factor market, Layard and Walters show that the price elasticity of demand a factor produced equal times share elasticity cost demand product final being product. Mathematically, = yV? , where e is the price elasticity of demand for the factor, , shrimp v is the share of the factor in the cost of the final product; is the price elasticity of demand for the final product, e.g. a shrimp entree a restaurant. absence adequate data on the retail market, the above relationship elasticity price might of demand elasticity used for the final of demand obtain indication product. for the final As long product price as v < is elastic Thi quite likely case relating shrimp final products can , the Obviously, the shrimp a shrimp dish is less than percent of the related food cost. Thus, 3441 percent represent upper bound proportion restaurant sales accounted for by shrimp. Thus, long greater the wholesale in absolute level value, price than elasticity percent of demand foi (approximately) shrimp that the retail level price elasticity of demand for shrimp products elastic. implication that demand shrimp elastic consistent with findings most previous studies shr imp market. However such a possibility provides some explanation for the contraintuitive results those previous studies. Recall comments Doll, quoted literature review, that one would expect consumers to be responsive to retail price. The Adeauacv of the Available Data as Proxies for the Desired Data Since required data exist, attempts estimate retail demand for shrimp associated elasticities must rely proxies for the desired data. Thus, the veracity of such estimates be addressed through a consideration of the suitability of the existing data as appropriate proxies for the desired data. Attention is focused retail food market shrimp prices (retail prices) proxies desired unavailable prices of shrimp institu tions (institutional prices). argument that retail prices are not good proxies for the institutional prices is based on a consideration probable production processes associated with products not use can retail outlets home consumption shrimp purchased in institu tions for immediate consumption are two distinct products. A difference in production processes argues against the existence a high degree the analysis of correlation of Gardner, assume between that t two product :he production prices. I process Following es in the two markets can be represented by the following production functions = f(s and I = f(s where R and are the quantities produced in the retail institu tional markets , s is the input of shrimp into processes , and a are bundles of marketing inputs specific to their associated proces ses. Gardner shows ratio a retail level price a farm level price can change differentially according source shock the market. example , Gardner derives the elasticity the retail/farm price ratio with respect a shift in demand for the retail product Epr/pf = NrSb(es eb)/D where N is the price the relative share elasticity input of demand for b in the price the retail pi of the retail :oduct product, elasticities supply inputs, function of N the elasticity of substitution between two inputs. Now, for the price in the retail market , to be highly corre lated with the price in the institutional market, Epr/Ps = NrSa(es ea)/D will highly correlated with Epi/Ps NiSb(es eb)/D, This are Ses, , Pr that elasticities of substitution between two inputs different in the two processes. elasticity substitution between two inputs retail market much closer zero see Heien) than institutional market where institutional operator have con siderable control over composition final product. retail market operator is relatively more restricted in the types amounts of marketing demonstrates that ; services markup pr can ricing to the shrimp behavior it sells. predictable Heien food retailers. Conversely, nature of the institutional product allows to be combined with a wider range of variable inputs such as other food ingredients labor than can the retail product. This implies that If the retail product the institutional product in fact two distinct products can be assumed that , Ni. Thus, shown that products are different, appears improbable that their prices are highly correlated are therefore not good proxies, one for the other. Empirical Evidence Relating to the Inadequacy of Available Proxies Some empirical evidence exists which lends credence to the notion that two products are distinct. Nash, in a study of purchasing patterns fresh frozen seafood, found that income a more pronounced effect on consumption of seafood away from home than it had on athome consumption. of the head of the household was also found have a significant effect on awayfromhome seafood consump are can are institutions, are different. Assuming both groups have access to both products, the finding of the product forms tha can they imply tend that consume either two product one or the other forms represent distinct products. Given a difference consuming groups difference products, plausible assume that one demand relationship may not be an adequate representative of the other. Keithly, in a study of the socioeconomic determinants at home seafood consumption, found that expenditures on meals consumed away from home were negatively related to home consumption of total seafood product forms with exception shellfish which insignificant, positive parameter. Assuming that some increase away from home food expenditures are for seafood, this finding imply that seafood consumed at home seafood consumed away from home are substitute products. Evidence a more general nature that food for home consumption food consumed away from home represent distinct products found in the work of Mincer. Mincer demonstrated that estimated income elasticities for a variety commodities will tend to be biased if the opportunity cost time is omitted. Evidence a similar vein found work in the suggests works that of Hiemstra the income Eklund elasticity in that of expenditures of Burk. for food Their away from home considerably higher perhaps twice high elasticity home food consumption expenditures (Prochaska Schrimper). Apparently addition preparation services Implications of Inadequate Data for Previous Studies further implication use wholesale level data proxy for retail data involves the prices complements or substitutes expected to be included in a demand function. Several previous authors have noted their inability (see Thompson et al., to demonstrate significant effects complementary substitute products demand for shrimp. lack correct data series explain these empirical problems. proper explanatory variables use input demand function are prices inputs outputs (McFadden, theory of retail demand holds that retail demand a function of the price product, income, prices other products (complements or substitutes). An attempt to identify complementary substitution effects are estimating to be implies a retail analysts demand consider function. function If the existing they data, e.g., U.S. correct shrimp data, this consumption fact data, explain are not adequate the expected proxies relationships are not obtained. indicate that previous attempts to estimate retail demand shrimp attempts estimate wholesale level derived demand which include theoretically correct explanatory variables should be viewed with caution. An example an attempt estimate retail level demand work Doll, used retail food outlet price data U.S. shrimp consumption (which wholesale level data) explain effect consumer price index meat, poultry, fish demand equation which employed wholesale prices standard wholesale level consumption figures. Since consumer price index should, theoretically an argument in a retail level demand equation while demand other two equation, variables should not belong remarkable wholesale that level consumer derived price index variable was found to be insignificant. A similar problem was noted Thompson et al., included producers stitutes tion, price in their employed index for processed (apparent) wholesale estimation level meat a price of retail prices level wholesale for shrimp shrimp level sub consump consump tion analysis. the wholesale level data are not adequate proxies for retail or institutional level data, their use in a demand estimation process would only appropriate estimation derived demand. fact that complementary substitute retail level products are considered appropriate arguments derived demand functions explain previous authors have experienced problems demonstrating complementary substitution effects in their demand equations. light these considerations, appears unlikely that retail market price of shrimp is an adequate proxy for the institution al market products price with because relatively two product distinct forms markets are probably production two distinct technologies. Further , it appears that previous efforts to estimate the retail level information on the prices and actual quantities of shrimp sold through institutions, seems appropriate consider estimate retail demand for shrimp associated elasticities as being only rough approximations actual values. Furthermore, with discontinuance collection retail prices probable difficulty (given of imputir diverse a price for shrimp technology) appears in the institutional that market calculation retail demand estimates shrimp U.S. become virtually impossible. Thus the model developed here will focus on the wholesale market level. Research Resource Constraints practical limits imposed research project resource con straints can also lead expediencies that cause mis specification. example, variables that could be considered endogenous have been entered into economic models as exogenous variables due to inadequate information available to the analyst Thompson et al.) as Sims observed, "because seriously explaining them would require extensive modeling effort areas away from the main interests of the model builders" Theoretical Constraints increasing understanding complexities economic processes , reflected in the work of Muth (1960 , 1961) Nerlove (1967) Lucas; Pierce; increased economists appreciation difficulty properly specifying many economic models , 6). prescribe econometric propel model r number (Nerlove, of lags 1972). or leads of Attempts ! variables in a account dynamic c known complex ities have created additional problems. For example, Sims notes including that create critique econometrics expectations identification standard text variables problems. econometric econometric Perhaps models Judge most model disconcerting contained authors recent note "that possibilities model misspecification are numerous false statistical models are most likely the rule rather than the exception" (Judge et al., p. 854). Policy Implications of Missoecification These critiques have significant relevance in the present context relating the economic impact of increasing shrimp imports likelihood development that existing policy econometric address this models will situation. be used Policies based misspecified models be inappropriate. For example the advisabil placing some restriction on the flow of shrimp into the U.S. linked to the demand for shrimp. Some economists, e.g., Prochaska Keithly , suggest the income elasticity of shrimp demand is high enough that, when combined with reasonable growth rate in U.S. disposable income the expected growth rate aquacultured shrimp supplies, barrier will nece ssary to prevent a decline in the level of real shrimp prices. implications research other economists are that trade indicating that shrimp can considered inferior good. Coupled with price inelasticity increased supply, an income elasticity the inelastic prevent range a decline implies in shrimp that barriers prices, since to entry growth in income cessary cannot expected to shift demand sufficiently to offset the depressive effects on price of increased supplies. Implications for the Present Effort To the extent that these contradictory implications are the result of misspecification of the econometric models used, a test system wide specification policy formation. may Given be useful in selecting the potential adverse a model impacts as the basis of of inappropriate policy, appears cation appropriate model used encourage policy the testing development. of the specifi Finally, development more accurate models of the shrimp market is to proceed from the base of existing knowledge contributed current models, would helpful some analysis an existing model would reveal those portions model most appropriate inclusion into respecification. Thus, appears that effort develop a more accurate understanding economic process, such operation market should begin with such a review existing work proceed with investigation the value newer techniques existing knowledge stock can knowledge then be used subject. in an attempt newly to respecify acquired traditional model development believe that follows testing the suggestion implications of Zellner structural Palm assumptions transfer functions and, add, final equations important element in the process of iterating in on a model that is reasonably accord with the information in the sample data. An obvious next step in the evolution of shrimp market models generalize the dynamic model used Adams allowing all right hand side variables to enter the model in appropriate degrees of lag. For example, lagged quantities have a significant effect on current quantities and, thereby, current price. This relationship seems especially relevant since such a large portion of shrimp flow through institutions. Institutions can expected have established capacity to market an expected quantity a given product that fluctuate around some trend. trend can seen the result the adj us tment process followed individual firms as they respond changes in their operating environment. One method of capturing the influence of this adj us tment process a variable interest lags variable predicting equations as Adams does. However such a procedure inadequate. assumes, method explicitly modelling implicitly, partial a theory adjustment of expectation processes formation (see Gould). As Maccin i has pointed out, schemes that relate expected variables solely past values the variable being forecast have been widely criticized because they assume that the firm "ignores business data, Maccini found that firms tend to utilize "economi cally rational" expectations making forecasts. "Economically rational" implies that firm depends on other relevant variables such as shifts government policy or other exogenous changes in the economic environment, addition lags variable being predicted, form expectations. These expectations turn, affect adjustment rates. level a single industry, such addressed this study, notion "economically rational" expectations imply that lags of variables other than the endogenous variable be useful explaining endogenous variable a given equation. Obvious candidates are lags of variables that appear in unlagged form. Thus, this study seeks expand previous efforts allowing more generous expression of lagged variables in what should be considered respecification of existing models of the U.S shrimp industry. Given acknowledged difficulties faced in the specification traditional econometric model, possible that an alternative, databased approach to model specification may a useful addition specification process. particular, initial vector auto regressive specification (VAR) the economic phenomenon under study useful since, argued Sims, can accommodate above mentioned modeling complexities considerable extent. Additionally, since a VAR model can seen unrestricted reduced form a generalized econometric model, parameter es large extent, this feature of the VAR model frees the analysis from constraint particular theory relating the operation economic phenomena under study and allows the sample data to speak for themselves. suggestion employ such databased techniques current interest in such techniques derive additional impetus from apparent exceeded economic ques fact advances theory. also that advances obtaining current be encouraged computational successful interest empirical in databased a recognition among capability models modeling economists have based techni that formulation, identification estimation dynamic econometric models must approached substantially new different ways (Sargent, 216). This study seeks follow spirit this search an integrated approach to economic model development attempt understand more fully operation U.S. shrimp market. CHAPTER MISSPECIFICATION TEST The value an econometric model estimated with two stage least squares (2SLS) three stage least squares (3SLS) procedures contingent on a number of conditions (Theil, 511513) two of which being that structural equations are correctly specified linear handle parameters nonlinearities, such variables. estimates Although exhibit methods reduced exist level efficiency, when compared to maximum likelihood estimators, unless specification linear variables. Proper specification par ticularly important a 3SLS analysis since misspecification single equation transmitted or "spread" across the entire system the 3SLS process (Judge et al., 617) Unfortunately, the probability of the correct specification being used appears to be quite low. An additional factor in support of this belief that misspecification tests of econometric models have been could, widely in the used. e past, This have failure been examine due to the lack systemwide necessary specification y statistical software computing power. Thus, it is probable that the results a test specification existing econometric model will indicate the model is misspecified. Note that term "misspecifica tion" appropriate since test used here searches ways which the existing specification inadequate rather than searching are __ purpose of testing an existing model of the U.S. shrimp market present time to demonstrate a specification test procedure that can more widely applied to understand strengths weaknesses of the existing model so that this knowledge can be included in a respecification of the model. analyzing tests existing be impractical. econometric For example, model without some detailed specification knowledge exis ting data resources impossible question particular regressors. Thus , in following analysis list endogenous predetermined variables not questioned, implying that the results of the analysis are contingent on the validity of the list selected the original analysts. Additionally, the functional form of the existing model must be taken as given, since alteration original specification would constitute a respecification model. Thus, the principle source of testable misspecification, once list of endogenous and predetermined variables functional form have been selected, set of exclusion restrictions placed on the model' parameters the analysts. set of exclusion restrictions placed on the structural model will imply restrictions derived or restricted reduced form the structural model. The veracity of these analystimposed restric tions can tested comparing the restricted reduced form with unrestricted reduced form of the same model. Because the unrestricted reduced form system equations with identical, predetermined use values consistent with population parameter values (Schmidt, 78). Since unrestricted reduced form not influenced exclusion restrictions, consistency of the reduced form estimates will not be dependent on the model specification. This result is not without some cost, as Dhrymes 127) has shown that the unrestricted reduced form estimators are asymptotically inefficient in comparison with 3SLS reduced form estimators developed from a correctly specified econometric model. The problem associated with the 3SLS estimator that structural parameter estimates, thus, their derived reduced form representation, will inconsistent structural equations are misspecified (Challen and Hagger 134). Hausman Test The relationships between the restricted unrestricted reduced form estimators motivate use of the Hausman test (Hausman) in the present analysis. test is based on the existence two estimators a vector statistics, viz. , vector parameter values simultaneous equation system's reduced form. Under the null hypothesis no misspecification of the model, one of the estimators, here, restricted reduced form parameter estimates derived from 3SLS estimation structural parameters , is asymptotically consistent efficient. Under alternative hypothesis, this estimator biased inconsistent. second estimator, here the OLS estimate unrestricted reduced form, consistent comparatively misspecified, two estimators will be dissimilar in value. Thus, the difference between two provides the data for the specification test. Vec(7o)  Vec(r3) difference between estimates o being the unrestricted reduced form estimator while is the restricted reduced form estimator. symbol "Vec( implies vectorization the matrix within parentheses (see Judge al., p. 949) Vec(iro) and Denote Vec(73) parameter covariance Then matrices a Hausman associated specification with test stati stic m = q' (var(q))l where var(q)  n3 the variance This test statistic distributed as chisquare with grees of freedom equal the number elements Note that if the system contains identitie only the reduced form parameters the behavioral equations would be used since including identities would cause and 0o to be singular. The covariance of the restricted reduced form parameter estimates is obtained a basis through discussion, a procedure assume outlined Schmidt the simultaneous 236239). equation model the form (4.1) YF +XA where Y and v are t x g matrices of endogenous variables and structural disturbances, X is x k matrix of predetermined variables, parameter matrices are respectively. implied, + v = o0 = Zp + v Note the restricted reduced form parameter estimates are functions structural parameters, i.e., AF"1 Thus, following result established the reduced form estimates can 385) be obtained variance/covariance matrix from (4.3) Var(ir3) = a~83/a8.'. B'r3/8a where is the variance/covariance matrix the structural parameter estimates (the elements derived 3SLS procedures. Schmidt provides a practical derivation of equation (4.3) (4.4) Var(73) = DW where, D=(r1)' W is a block diagonal matrix with , i= 1 , given The matrices are the and ki endogenous and predetermined regr essors appearing in the ith equation. In practice columns the estimated reduced form parameter matrix are used first columns since plim(X'X)X'Yi converges vector population parameter values associated with endogenous variables equation. remainder With submatrix auxiliary regression predetermined variables appearing equation complete regres sor matrix. variance unrestricted reduced form Var(7o), obtained from a seemingly unrelated (SUR) estimation of the unres tric ted reduced form. This procedure allows the crossequation covariances to enter test. The variance/covariance matrix of the SUR parameter estimates can be represented W' D' Wi=plim(X'X)1X'(YiXi) where e X) 2 is the error covariance matrix from the OLS estimation the unrestricted reduced form parameters, i.e. (YX(X'X) 1X'Y) YX(X'X) X'Y) The Structural Model structural model seven equation, simultaneous equation model U.S. shrimp industry based monthly data (Thompson, Roberts, Pawlyk). This model was briefly discussed in the preced chapter. The data for reestimating the model were kindly provided Dr. Kenneth predetermined equation Roberts. seven overidentified Counting endogenous with the intercept, there variables total are fourteen model. number Each of overidentifying restrictions being fifty. The endogenous variables in the model Consumption warehouses) (disappearances thousands from wholesale of pounds Wholesale price 2630 count frozen shrimp, York ($/lb. Exve ssel Mexico price ($/lb.) month of 2630 thousands cold count shrimp, of pounds, storage Northern Gulf lagged one (stocks), month thousands pounds Imports, thousands of pounds Landings pounds from U.S. Gulf of Mexico ports thousands Fishing effort, number fishing trips Gulf shrimpers. The predetermined variables in the model Wholesale price, lagged one month are are month cold storage holdings (stocks), lagged month Currency exchange rate between U.S. Japan (yen/dollar) lagged two months Unadjusted retail sales in eating places Prime rate of interest on shortterm, business loans Diesel fuel price (dollars/gallon) Avera ge precipitation in coastal Louisiana, inches lagged two months Avera ge atmospheric temperature coastal Louisiana (degrees Fahrenheit) lagged two months Quarterly dummy variable for second, third, and fourth quarter of the year (j=2 ,3,4) Error terms (g=l, ,3,4,5 All equations form as are linear in the parameters are shown in functional = f(Pw = f(Sl Pel,Q , Q4 = f(L, = f(S1 = f(Pw2 Pwl, Q2 , p4) , Q2 = f(PR2 , Q2 , Q4, = f(Pe p7). authors provide very little theoretical justification of the model specification , in general, not specifically identify behavior being modeled each equation. Thus difficult provide m iuch more information re arding he rationale ol their model one . In discussing their specification, the authors note that economic theory suggests that prices substitute complementary products be included in their first equation. However, their prelimi nary index tests to identify for processed substitution meat were effects unsuccessful using and t the producers therefore ' price the variable was not included in the model. Also, citing the inability of previous researchers to demonstrate significant complementarity or substitution effects the authors did not include ices of potential substitute or complementary products. Thompson et al. describe the second equation in their model as a price level exvessel equation, prices noting motivates that this the inclusion designation current assures that lagged wholesale exves sel prices move together. dependence of exvessel prices on market conditions wholesale level treated including current lagged wholesale price exvessel price equation. Noting that the United States Japan are the major competitors world supplies shrimp, authors include rate exchange between the Japanese yen and the U.S. dollar in the equation explaining imports of shrimp into the U.S. The equation explaining landings of shrimp reflects influence of environmental factors on the annual shrimp crop/population. quantity harvest landed is dependent extent on the fishing amount of shrimp effort available exerted industry. account pink life Three for 98 shrimp cycle. cies percent are estuary Growth r of annual dependent :ates shrimp Gulf during (brown, landings. during this the early time white, Brown, stages l be will and pink) white, and of their adversely r r  w I Louisiana wetlands. was used Data reflect precipitation salinity were used levels instead Louisiana salinity levels monthly because basis latter Average were atmospheric consistently temperature available was used instead water available temperature in in a consistent estuaries timely fashion. latter Shrimp was landings follow typically and July a seasonal low from pattern January reflecting from through spring year April season year. id high brown Landings in May, shrimp are June, gradual season dummy decline for white variables in landings shrimp which after July reaches are included is interrupted a peak equation in October. to reflect the fall Quarter seasonal  in landings (Thompson et al. , p. authors note that treat production response industry adequately it is necessary to include the equation explaining effort expended harvesting shrimp. number shrimping trips made industry vessels selected proxy effort. authors explain that the existence of externalities in the shrimp shrimp fishery landed. increased "Hence, effort or may equations not were increase included amount to describe 'behavior' industry terms landings effort, respec tively" (Thompson et al., Hausman Test Results Discussion The structural model was estimated 3SLS. The results were total agreement with those published Thompson uncertainty regarding exact form several exogenous variables. However, since same data were used to construct both the restricted the unrestricted reduced forms and their assoc iated variances, this discrepancy will affect test results. Following steps outlined previous section, Hausman's statistic not was the critical value for a chisquare variable with 98 degrees of freedom confidence level, null hypothesis that model correctly specified is rejected. itself, Hausman test result limited value in dis covering possible causes for the rejection of the null hypothesis possibly finding avenues improving specification econometric model. Some desired information can obtained from a consideration two sets parameter estimates. facilitate the comparison of the 3SLS reduced form parameters from restricted model with the OLS parameters estimates from the unrestric ted reduced form, Table presents both sets parameters along with the tvalues for each parameter. A large difference between estimated parameter values coupled with an indication parameter significance as shown the associated values greater than two signals a possible source of model misspecifi cation. obvious example fuel price variable which designated letter In six seven equations the OLS estimate is much larger in absolute value than the 3SLS estimate. consistency of plausible signs on the fuel 1 price variables consis tent indication significance across seven equations argues inclusion of the fuel price variable more than equation a respecification of the model. Apparently, this variable reflect the fact that changes in fuel ces have impacts at all market levels. Further, appears that fuel a substantial effect one Table Comparing Restricted Reduced Form Parameter Estimates with Unrestricted Reduced Form Parameter Estimates Description of Symbols Symbol Description Symbol Description Restricted Reduced Form Unrestricted Reduced Form Constant Second Quarter Dunmy Variable Third Quarter Dunmiy Variable Fourth Quarter Dunny Variable Expenditures in Eating Places Interest Rate Fuel Pric Exvessel Cold Stor Wholesale Wholesale Yen/Dolla Precipita Temperatu e Index Price Lagged One Period age Lagged One Period Price Lagged One Perio Price Lagged Two Perio r Rate Lagged Two Perio tion Lagged Two Periods re Lagged Two Periods APPARENT CONSUMPTION EQUATI Parameters RRF URF Cn 15931.17 21872.27 02 2406.30 463.02 6777. 6666. 1. 21. 3. 793. 0. 56. 277. 2. 1. 1. 12234. 1757. 0. 817. 779. 69. 54. 217. ON tValues tRRF tURF 9.59 2.12 1.81 0.32 4.98 1.62 5.04 0.08 6.03 6.69 0,94 0.21 0.42 2.08 1.15 0.63 1.00 0.31 0.48 0.34 1.13 0.46 1.07 3.57 0.54 0.34 0.59 2.99 EXVESSEL PRICE EQUATION Parameters RRF URF Cn 1.87 2.00 Q2 0.16 0.09 Q3 0.36 0.17 Q4 0.10 0.00 E 0.00 0.00 R 0.05 0.04 F 0.01 0.79 Pel 1.03 0.79 S1 0.00 0.00 Pwl 0.12 0.14 Pw2 0.36 0.05 EX2 0.00 0.00 PR2 0.00 0.01 T2 0.00 0.00 t tRRF 2.88 1.32 2.37 0.72 2.28 2.74 0.44 3.70 2.37 0.57 2.50 1.90 0.61 0.67 Values t 4 1 1 0 1 3 2 6 3 1 0 3 1 0 LANDINGS EQUATION Parameters RRF URF Cn 20957.91 25404. Q2 3776.62 4210. Q3 1841.08 3395. Q4 2206.48 3913. E 0.08 2. R 20.70 161. F 1038.65 11318. Pel 457.52 2687. tValues tRRF tUJRF 3.47 2.84 1.86 3.34 0.57 1.73 0.82 2.26 0.71 3.21 0.72 0.72 0.58 2.22 0.73 1.12 0.76 0.62 0.45 0.16 n 70 A c WHOLESALE PRICE EQUATION Parameters RRF URF Cn 2.13 2.60 Q2 0.16 0.10 Q3 0.41 0.18 Q4 0.14 0.00 E 0.00 0.00 R 0.03 0.02 F 0.00 0.79 Pel 1.17 0.44 S1 0.00 0.00 Pwl 0.08 0.51 'D,.I n t.i n n tValues tRRF 3.14 1.22 2.74 1.00 2.48 1.76 0.44 5.83 2.36 0.53 4, Tl d ds ds Table Continued: COLD STORAGE EQUATION Parameters RRF URF Cn 27353.80 25497. Q2 502.80 431. Q3 2067.62 1120. Q4 80.54 541. E 1.34 0. R 31.73 134. F 829.51 4942. Pel 927.49 1650. S1 0.97 0. Pwl 83.37 115. Pw2 1425.36 177. EX2 11.32 13. PR2 266.06 215. T2 389.93 330. tValues tRRF tURF 5.08 2.99 0.30 0.36 0.78 0.60 0.04 0.33 4.15 0.51 1.06 0.63 0.58 1.02 1.32 0.72 57.07 25.37 0.50 0.06 3.42 0.13 1.55 0.82 1.38 1.64 4.66 5.52 IMPORTS EQUATION Parameters RRF URF Cn 7584.98 16969 Q2 958.78 3334 Q3 1462.16 4874 Q4 7483.64 3546 E 0.00 3 R 0.00 233 F 0.00 8162 Pel 0.00 1382 S1 0.00 0 Pwl 0.00 1386 Pw2 2046.29 924 EX2 16.25 68 PR2 0.00 56 T2 0.00 86 tValues tRRF tURF 2.47 1.92 0.84 2.67 1.27 2.50 6.64 2.07 0.00 4.79 0.00 1.05 0.00 1.62 0.00 0.58 0.00 0.04 0.00 0.67 6.58 0.64 1.78 4.11 0.00 0.41 0.00 1.39 FISHING TRIPS EQ Parameters RRF Cn 58747.00  Q2 13132.00 Q3 3093.84  Q4 8091.91 E 0.38 NATION URF 47905. 15391. 11584. 9508. 0. 388. 3481. 337. tVa tRRF 3.45 2.09 0.34 1.00 0.88 0.91 0.67 0.94 0.94 0.49 0.91 0.86 1.36 4.06 associated point that since structural model only permits fuel price to enter trips equation, appears that this variable' s effect not communicated rest restricted reduced form equations. This suggests that trips equation is not The explanatory weakness of the trips equation be due to problems inherent in the trips variable. For example, a proxy the trips fishing variable, effort, included in the structural inappropriate this model task. Shrimpers take same number trips time period, but adjust amount fishing time trip on the basis fuel shrimp prices. Thus, the number of trips they make not be very powerful in explaining fishing effort. These considerations argue for dropping trips equation from a respecification of the model adding fuel variable in several of the other equations. Another differences "expenditures variable between eating associated restricted places, with d and " E. significant parameter unrestricted reduced Since the OLS estimate forms estimates parameter are often larger than those estimated 3SLS estimator be argued that the econometric specification results impact especially econometric apparent expenditures important model's consumption being considering estimation on imports. underestimated. policy impact This implications expenditures The unrestricted model suggests that expenditures have a much larger impact on landings as well on consumption positive Thompson on imports side, than appears decision suggested the unrestricted exclude expenditures econometric model from model. supports wholesale price, exvessel price cold storage equations. be used calculate an income elasticity of demand shrimp. their article, Thompson calculated elasticity demand related retail expenditures inelastic range (.42 percent). However if the unrestricted model correct in indicating that econometric model substantially underestimates the impact expenditures consumption, then possible that income elasticity demand for shrimp is in the elastic range. As shown Table 4.1, the unrestricted reduced form parameter estimate on expendi ture three times larger than the restricted reduced form estimate. This into difference the elastic in value range. enough The policy to boost implications estimated elasticity one estimate versus the other are substantially different. The results relating to the Yen/Dollar exchange rate variable also merit discussion. The unrestricted model indicates that this variable should added apparent consumption equation. In agreement with model indicates specified exchange Thompson rate variable should unrestricted not model included wholesale tion price equation of the tvalues or the exvessel listed in Table price indicates equation. that An examina the two models largely not completely agreement regarding the restrictions placed on the model the analysts. final point relation between reduced form Hausman test developed here structural test Hausman presents in his paper. This latter test compares Vec(f3) Vec(8 2) where these are al., the off diagonal elements of the error covariance matrix are near zero. Additionally, there no guarantee that of the elements of $2 consistent if the overidentifying restrictions are incorrect. test this type, i.e., comparing Vec( 3) Vec( 2) Thompson model. results indicate that the hypothesis proper cancer specification level. cannot The inconsistency rejected of the t conventional wo tests an obvious signifi subj ect for further research. Although specification test results relating the existing SEM indicate that the null hypothesis no misspecification could rejected, analysis reduced form parameter values generally supported, with some notable exceptions, original analysts' theoretical decisions regarding which variables to include a given equation. in the existing These 4 may results not be may very indicate serious. that the misspecification Indeed , a comparison policy implications existing model with those respecified model to be discussed in Chapters and VII suggests that policy implications existing model are robust misspecification. However, results indicate that several variables should disagreement with be added to the model the theorists' in various parameter locations, restrictions. indicating These results should be useful in respecifying the structural model. are was run Summary CHAPTER VECTOR AUTOREGRESSIVE MODELS primary purpose this study better understand relationships among relevant economic variables associated with U.S. shrimp market t for policy making purposes. Typically the approach to this metric problem model SEM). sought within However, context several a simultaneous reasons, econo including analysts' among incomplete the variables understanding of interest, time the resulting related simultaneous relationships models are misspecified. This study hypothesizes that the analysts understanding of the economic phenomena under study the analysts' later specifi cation a related can be improved the knowledge gained from specification analysis a vector autoregress (VAR) specification of the subject. Vector Autoreeressive Theory Understanding the reasoning for selecting the VAR approach begins with standard explanation that the simultaneous econometric model expressed part (the complete system contain trend component: see Judge et al., P. 686) of a multivariate, autoregressive moving average time series (ARMA) model . Quenouille, 1957 Theil Boot, 1962 Zellner Palm, 1974; Wallis , 1977 or Anderson al. , 1983). A general dynamic simultaneous equation model (SEM) be written can can A(L)Zt + B(L)Xt = C(L)et D(L)Xt = F(L)vt, where F(L) a vector are matrices of endogenous of polynomials variables , A(L) in the nonnegative D(L), powers of the operator are vectors a vector of random exogenous serially variables, uncorrelated errors where, assumed have zero mean constant variances. system combined to form the single ARMA model (5.2) G(L) H(L)pt where G(L) H(L) are appropriately restricted matrices having nonzero is the elements vector A(L) , B(L) containing C(L) both and Xt. respectively In general, J(L) = JO + J1L + J2L2 + J3L3 where = A, etc. H(L) invertible, ARMA model can expressed purely autoregressive form (5.3) A(L) where A(L) been redefined A(L) G(L) Then, autoregressive form in ( 5.3) can be written a generalized, restric reduced form = A(L)Yt_ where A(L) has again been appropriately redefined. The equation system (5.4) is restricted such that the Xt subvector of Yt is exogenous. If these restrictions are not imposed the unrestricted reduced form can = #t, ted, , B(L), C(L), D(L), F(L), =(H(L))1 known vector autoregressive systems. Thus, system unrestricted reduced form some unknown structural system equa tions; alternatively the VAR model is consistent with many different structural models. These considerations argue in favor of using VAR specification of the economic phenomena under study because specification minimizes exposure the recognized dangers analyst observe induced relation specification to univariate errors. times As Anderson series et al. estimation: 250) "the time series analyst not constrained particular theory during identification and estimation of the model second argument use specification arises directly ted. from Note the generalized that nature generalized of the VAR reduced system form just equation demonstra system equations have identical regressors long restrictions imposed. Thus, the system in can be consistently estimated using ordinary least squares without being concerned about the existence simultaneous equations bias. Furthermore, estimating each equation separately using ordinary least squares produces asymptotically consistent estimates because the righthand side variables are the same every equation. From a practical point of view, then, models are easy to estimate because efficient consistent estimates produced without using system estimation procedures (Hakkio and Morris, 10). More important, however are implications this consistency estimation desired population parameter use are can SEM, then the estimates of properly specified model are consistent with the information in the data sample therefore, consistent with underlying, but unspecified SEM. Thus, model provides practical method obtaining consistent estimates population parameters use in the prediction and control of economic phenomena, while minimizing exposure specification errors made in a limited information environment. Some preliminary evidence relating to the robustness to misspeci fiction models versus SEM models been presented Hakkio and Morris (1984, Their results indicate that point estimates from a VAR model are more robust to model misspecification than point estimates estimates from from a structural a structural model. model However, are more they efficient found than that those point from a model except when model badly misspecified. Hakkio Morris speculate that specification more robust because a less restrictive system of equations than is the SEM. additional attraction specification that parameter matrix, the matrix, A(L) , in equation system contains information on the relationships between the variables included in the vector This information can quite useful prediction purposes can also provide guidance in policy matters. Thus, specification several characteristics which recommend in applied work. Estimation Procedures use above, this results least squares parameter estimates coinciding with the maximum likelihood estimates. Once exogenous endogen ous variables to be included in the model are selected, the specifica tion the model is completed deciding where the variables enter model. model composed three parts exogenous variables, endogenous variables, error vector. exogenous variables will include the intercept, the indicator or dummy variables, trend exogenous system variable, of endogenous other variables. variable Such judged a model represented (5.5) = bt + A(L)Yt where s a vector of the endogenous variables a vector composed the indicator variables the trend and intercept variables, the selected exogenous variables. Determining the Order of the Model A complication ar ises in selecting the order of the autoregressive portion of the model. This selection accomplished sequentially adding lags of the vector to the model testing whether the added variables have estimated parameters that are significantly different from zero. Several tests of the null hypothesis that parameters of the added variables are not significantly different from zero are avail able. Nickelsburg studied the smallsample properties of six criteria using Monte Carlo techniques found none to be clearly superior can several criteria base decision combined information. Accordingly, four criteria were selected for calculation. The criteria Akaike s AIC criterion, Schwartz s SC criterion, likelihood ratio statistic (LR), and Sims modified likelihood ratio test (SMLR). The criteria are represe nted AIC(j) j)/t SC(j) = In j = In i2j Int)/t, LR(j m)= t(lnI lnlSm SMLR(jm) t((k*m)+s )(lnlZj lnlEm where is the error variance/covariance matrix of the test version the model having j included lags of the dependant variable vector, k is the dimension test of the version vector model, m are the number s is of lags number in the exogenous regressors in the two criteria tem (Sims select The decision the model of order rule that minimizes for the first the value of the criterion. The last two criteria are distributed asymptotically as chisquare with degrees of freedom equal the number of restric tions placed on the restricted model k2(mj). Thus the decision rule test, pairwise models increasing order select the model with the largest not rejected as insignificant. purpose adding lags dependent variable until error process becomes white noise remove influence past values variable present value. This process accordance with the notion that a major portion of the variation are + (k is also required in Chow' technique , discussed below, of analyzing lead/lag relationships among variables the endogenous system. The variation remaining error process after prewhitening will a combination of random variation the variation caused other variables . Greenberg Webster, 153). Thus, analyz residuals of prewhitened time series influence of other variables on a variable interest .can be detected. technique employed here analogous to the prewhitening processes employed wellknown causality testing procedures however, instead pre whitening each time series individually vector time series variables are treated simultaneously. Hoskina s Test for White Noise Testing a white noise error process context multivariate system accomplished using Hosking s (p. 605) multi variate portmanteau statistic tr(Cr'Co = Zr=1 .. 1CrCo where n is the number of observations , tr is the trace operator, are the estimated autocovariance matrices of the vector of endogen variables zero levels, respectively s=o(nl/ Note that s=o(nl/2) implies that s is at most of order n1/ (Greenberg Webster, 313). portmanteau statistic dis tribute as chisquare with k2(sp) degrees of freedom, where k is the dimension endogenous vector order process. ous Estimating Relationships Among Variables Once the VAR model has been estimated, the information concerning the relationships two alternative parameter matrix among the variables techniques. of the simultaneous in the model technique system involves to discover can be obtained decomposing the trigonomet c functions implied its characteristic roots. The other detects relationships between two variables considering correlations between variables various distances time, i.e. various levels. These two techniques are also known analyses frequency dimension and in the time dimension, respectively. techniques will be discussed sequentially. Decomposition of the Parameter Matrix exogenous variables in section of the model at ( can be dropped since they are hypothesized to have no simultaneous relation ship with endogenous variables simultaneous system. generalized model becomes equivalent model (5.4) which can rewritten without the lag operator = AlYt1 + A2Yt2 *+* YAmYtm + Ut. adding an appropriate identity for mi of the m lags this system can be rewritten S Am * a O 0 0 Yt4 two redefining vector names, system can be written as a first order system with the more compact matrix form = AYt1 + Ut. Here are vectors, being number variables in each vector and m being the number of lags and A is the km x km matrix parameters. Since error vector would have been shown insignificantly different from a vector zeros estimation testing phase discussed above, can now set equal zero the equation system in ( becomes a homogeneous , linear system of difference equations of the form, = AYt1. The solution to such a system is Yt = AtY0. Assuming quickly moment appreciate that characterizes are scalars time one can path following relations can seen if I if I the solution the solution path path is explosive is damped the solution oscillates the solution path is monotonic. case a multivariate system, in which regain their path matrix of Yt. designations, course, the matrix eliciting A again the desired characterize information time is somewhat more complicated under these more interesting circumstances. Chow (chpt. shows this is accomplished utilizing the characteris roots characteristic vectors of matrix A. Freauency Dimension Analysis A characteristic root of A is a scalar, that satisfies A II= With A being a square matrix of dimension where p=km, roots will be obtained. Associated with each characteristic root, characteristic vector, which satisfies Aibi. Writing this equation in expanded form yields A(bI , bp) = (Albl, , bp) Denoting matrix , *bp) whose columns characteristic with vectors the characteristic A and roots setting equal represented to the diagonal along its main matrix diagon system of characteristic roots vectors can be written AB = equivalently = BDB1 This implies the solution, = AtYo be written = BDtBlYo. simplify some mathematical computations involved deriving this solution, Chow, introduces the canonical variables, with vector form representation: B1Yt = BZt. Using the expressions = BDB , Ztl = AYt_1, redefined B1(BDB DZt1. solution to Z t = DZt is Zt = DtZ0. Thus, the solution for Yt can be expressed terms canonical variables BDtZo. solution for individual elements of Yt is given = B;P iZlnti) *1 nto) + + B (Z*n t are can can , pbp) = (bI = BYtl 1)Ytl v ^ * / f r^ * Y *n indicated. Further analysis these complex numbers can reveal information concerning the related variable's cycle frequency phase shift (lead or A complex lag) root, relationship with is defined other variables as A in the = a + bi where system. 1)1/2 a and are real numbers. term, complex, arises from the fact that the complex root is composed a real part, and an imaginary part, being imaginary number. complex number, represented on a Cartesian plane with imaginary values mapped along vertical axis real numbers along the horizontal axis. The length a ray from complex the origin number, to the point d is called (a,b) the absolute the modulus value Using plane trigonometry, modulus calculated , cosO or a cos8 sine where is the angle between modulus horizontal axis. Using these results complex characteristic root can be expressed as A =a +bi (cos0 *sinO) Xlei0 The relation, = cos8 *sin0 used here explained by Chiang . 518520). Returning the solution an individual time series variable given above in equation consider the contribution a pair of complex conjugate roots, say, would make the time path of Yit Note that cost *sin t) De Moivre' theorem (Chiang, 522). Thus t(cos~t + i sin t) = Rtei t(coset  i'sin0t) = Rte i0t can (a2+b2)1/2 pairs complex conjugates such that BIZ10 then B1(ZIOXtl) from equation can repre sented by SRtei e + SR ist = SRt( ilt) = SRt cos( +0t) SRtei e it + isin(#+Bt) + SRte cos(4+Ot) it sin(+Bt) SRtcos(t+Ot). Thus the contribution a pair of complex conjugates a cosine function time multiplied a factor, As with the previous discussion relating to the characterization a solution a univari ate system, this time path will be damped or explosive depending on value of R Note that the complete solution of Yt is the sum of the various individual contributions as shown equation involve additional pairs complex roots singular, real roots. studying the expression, 2SRtcos (+Bt) one can understand relationship between various time series (variables interest) determined. Note that if 2SRt = 0, equation would simple cosine function time that would repeat itself every time Ot became some multiple of 27r. For example, cost when etc. More generally cost when 27r+k, 47+k, etc. Dividing both sides of this last expression reveals that cost will have same value when = k/O, 2r+k/6 4ir+k/9 every = 2Ir/O time units. Thus , the time length the cycle is 2r/8, where is measured in radians. Equivalently, one cycle is completed 2/eO time units. frequency function number can , 2ir, = k, B2Z20 eie because there will a frequency corresponding to each pair of complex roots in the solution of Yt nonzero, the value of cos(*+0t) will be shifted forward or backward according value For example , assume one time series obeyed the function cos(Ot) at frequency, a second time series have followed cos(+Bt) the value at the one when same while frequency; the second the first series series would would have value one when = /e0. second series arrives at or becomes one before the first series. Equivalently, the second series leads first series time lags series time the second units. time Stated units. Thus, differently is the phase first shift in number time units function, cos(4+et), indicates leading relationship with cost if $ is positive or a lagged relation ship if is negative. If both time series have nonzero the lag or lead relationship between two, at a given frequency, is given by r(sr)ij = abs(ji'j)/8r. Conventional usage is that if Oi>ej, series leads series r(sr)ij time units at frequency, However , when r(sr)ij is subtracted from a complete cycle leadlag relationship reversed. This result reflects cyclical nature underlying trigonometric functions implies that causal hypotheses take form, leads Yj t, form, leads appropriate conclusion relating leadlag relationship depends results If , 2n/6, summary, contribution a pair complex roots time path solution a times series variable can obtained calculating an expression of the form, 2SRtcos(4+Ot). The expression, 2SRt will define the amplitude of the function, e.g., the magnitude the swings above below trend line. The function will tend converge trend line if R<1 tend to explode diverge length from (will trend complete line a full if R>1. cycle) The in 2/0B function time will units have which ia cycle implies a cycle frequency of 9/2w. The phase of the cycle will be defined time distance between two series will defined r(6r)ij abs( i Finally, leadlag relationship between two series establish ed on the basis of hypothesis testing. The estimated dynamic properties of the various series in the VAR, as determined parameters of equation (5.7), are functions of the parameters submatrices, , from equation Thus, variances of the dynamic parameters are related to the variances of the parameters matrices. 385) shown that if Y f(X) then the variance of Y is var(Y) (BY/aX) ' cov(X) *(aY/aX). present case , the phase angles frequency angles are functions elements matrix equation i.e. , they function parameter estimates. Thus, variances estimated angles and frequencies can be found using s result. example the variance of the frequency angle var(9) = (80/aA) *cov(A) *ae/aA, out are 4j)/ r. differentiation, i.e., adding a small value to individual elements dividing change change individual parameter estimate. The variances of the frequency angles can be employed in testing hypotheses that individual frequency angles are not significantly different from zero. Assuming the frequency angle values are normally distributed, z statistics, where = (60)/a and a  the square root the variance of the frequency angle, can be calculated used in the hypothesis tests. Since these tests are valid only sample stati stics are normally distributed, a test of the normality assumption should be conducted prior to testing. Time Dimension Analysis A second method of determining the relationships among variables of interest to analyze the variables of interest are correlated over time. This procedure is facilitated considering group variables interest vector. correlations can then calculated autocorrelations, i.e., correlation variable with sequentially higher ordered lags of itself. Using matrix algebra, result will a series matrices whose elements will covariance variables relationships of interest arn among variables e the variables interest. in the endogenous Here vector of the vector autoregressive model. autocorrelation matrices were calculated following procedure outlined Chow 4954). Subtracting the expectation where tE(Y Multiple substitutions into equation replacing yields (5.10) = Ut + AU.t. + A2Ut_2 +At 1Ut. The autocovariance matrix, for r > 0 E(Y* tr) = (Ut + AUt + A2Ut.. ... +AtlUt) (Utr + AUtr1 r1Ut) + A2Utr2 tr) = (ArUtrUt r + Ar+lUt r1Ut rl ... + At1U1Ut r1 'A',trl) = Ar(V + AVA + A2VA ... + Atr1VA'tr where E(UtU for t/s Note that current period variance/covariance matrix (when r=O) the matrix inside paren theses equation Denote variance/covariance matrix with the symbol practice this matrix is determined by setting such that AnVAn' negligible. example present analysis was set to 200. autocovariance matrices relating covariances between error terms separated various lengths time are obtained through equation this raising sequentially, the value arranging r and the resulting recalculating. matrices doing in increasing order one can observe covariance between two times series changes as one variable is compared to larger lags of another variable. covariances between two variables, also termed cross covariances, reveal lead/lag relationships between the variables. example, value crosscovariance between two variables +At  E(UtUs)=Vt correlation between two variables obtained dividing crosscovariance square root of the product of the variances two series. The variances of each series are found along the main diagonal of the variance/covariance matrix (where r=0). Understanding significance calculated correlations requires measures standard errors of the correlations. These statistics are obtained using technique, based on Rao result, described earlier. In this case, the variance/covariance matrix of the autocorrelations, var(O) = (8a/aA) '*cov(A) *8a0/A, where cov(A) variance/covariance matrix estimated parameters. value , an/aA, is estimated numerical differentia tion, i.e. , by sequentially adding a small value to individual elements dividing change in n change individual parameter estimate. practical aspects this process are somewhat complicated however, is possible to provide the basis an understanding process with a short example. Assume a model wherein the number estimated variables parameters in the model equals equals number Thus, autocorrelatioa endogenous n matrices at each are symmetric with dimensions Also, assume that the variances of autocorrelations out to a lag are desired. process small would .000001) to change amount value and recalculate one estimated the autocovariances parameter out a lag numerical derivatives associated with change a particular parameter estimate. Because there are parameter estimates there will be 50 matrices of derivatives. Next matrices of derivatives must sorted so that derivatives associated with a given order are same matrix. Next, each of these 11 rearranged matrices with dimensions sorted into smaller matrices with dimensions 50 by that contain only derivatives associated with a particular endogenous variable assumed (recall here). that Each there these are endogenous 55 rearranged variables matrices model are used post parameters multiply to obtain variance/covariance the variance/covariance matrix matrix estimated of the autocorrela tions at each order of lag. variances of the correlations between a single endogenous variable itself the other four variables each order of lag, are located along the main diagonals of the 55 resulting matrices. vertically Extracting each concatenating of these whole group diagonals result as a row vector a 55 matrix containing 11 submatrices. Each the 11 submatrices contains variances autocovariances between five endogenous variables Once at orders of lag variance from zero matrix to ten. of the autocorrelations is calculated can be used to test the hypotheses that the estimated autocorrelations significantly different from zero. complication arises calculating test statistics such zvalues, where (c0)/a are are normally distributed. They lie within the unit circle. Thus appropriate transformation of the correlation values is required before hypothesis tests based normal distribution can used. Assuming transformed autocorrelations are normally distributed implies that a z statistic can be calculated. If the z statistic greater than two, hypothesis on the autocorrelation value rejected. This chapter estimation, discussed the frequency vector time domain autoregressive analyses model, for estimating relations chapter between provides the endogenous empirical variables applications of the model. of this following material. can CHAPTER ESTIMATING VAR MODEL Estimating a vector autoregressive model of the U.S. shrimp market entails selection variables included the model, selection order the model, testing assure that residuals discussed model sequence. are time The chapter independent. concludes These with steps a section will discussing the estimated the endogenous parameters variables of the VAR as revealed model through the relationships analyses among in the frequency and time dimensions. Selection of Variables in the VAR Model selection vated of variables understanding included market the model probable is moti effects changes important variables will have on market participants. additional exvessel selection price criterion U.S. shrimp parsimony. Of major openentry concern nature shrimp single existing fishing price v a data industry. ariable U.S. However in the model shrimp uti wholes al because lization price was it relates chosen most importation. directly Further, exvessel prices can be derived from the wholesale price series Imports of shrimp were included reflect increasing importance this variable market. Domestic landings are added other source of shrimp supply to the U.S market. Endofthemonth cold storage holdings shrimp in U.S. ware houses were behavior; included however, they primarily y are also capture essential effects for calculating inventory an estimate apparent consumption. Thus the inclusion cold storage holdings obviates estimate need apparent to include a] t consumption parent can be consumption, derived once since an adequate landings, imports, and cold storage holdings are estimated. Another maj or force in the market thought to be the Japanese yen/U .S.dollar exchange rate (Thompson et al.) since foreign suppliers choose either these two markets depending relative strengths reflect of the a weakening two currencies. of the An increase relative in the yen/$ to the dollar ratio thus would foreign suppliers will find sales in the U.S. relatively more attractive than sales to Japanese buyers. Theoretical considerations suggest the inclusion of income prices substitute commodities. Following Thompson al., U.S. aggregate expenditures in commercial eating places is used as a proxy for an actual income variable such as U.S aggregate disposable income. "expenditures" justified because approximately U.S. shrimp consumption occurs a commercial setting. Prices substitute products are included in the model dividing the wholesale price of shrimp expenditures gross national product implicit price deflator. The price deflator can be considered an index of all other input prices thus, inclusion brings prices use rational economic agents only respond changes real prices. Completing the list of variables in the model are a trend variable eleven (monthly) indicator or dummy variables. These variables were included to account for temporal persistence and seasonal effects. Selecting the Order of the VAR Model Once specification variables t< of the model o be included completed the model are deciding where selected the variables enter ogenous model. variables model , the endogenous composed variables, three and the parts error vector. exogenous variables will include intercept, indicator vari ables, trend variable, other variables judged exogenous to the system of endogenous variables. In the present model, the expenditures variable was judged to be exogenous. remaining variables (landings , imports, wholesale price, cold storage holdings, the yen/$ ratio) enter the model as endogenous factors. The model, as shown at 5 Chapter can be represented t = bt + AY.t_ where a vector endogenous variables and b a vector composed of the indicator variables the trend and intercept variables, the expenditures variable. Estimation model accomplished using ordinary least squares techniques. actual calculations were made a micro computer using program displayed in Appendix program written matrixbased programming language entitled "Gauss ex As described in the previous chapter, this selection is accomplished by sequentially adding lags vector model testing whether added variables have estimated parameters that are sig nificantly different from zero. noted the previous chapter, the criteria selected with order test are Akaike criterion, Schwartz' s SC cri terion, likelihood ratio statistic (LR), Sims' modified likelihood ratio test (SMLR). decision rule first criteria to select the model of order that minimizes the value the criterion. The last two criteria are distributed asymptotically chisquare with degrees of freedom equal to the number of restrictions placed restricted model, symbolized k2(mj). Thus, decision select rule the model test with , pairwise, the largest models lag not increasing rejected order as insignificant. Table presents results of calculating values four lagged criteria dependent shrimp vectors. model Note over that increasing AIC, numbers SMLR added criteria indicate the model with should be selected. The degrees of freedom between rejecting two adjacent the null models hypothesis Thus 37.653 , the critical percent value confidence level. The SMLR would rej ect all models of order higher than one until twelfth order. The values of the maximum likelihood ratio test, given column headed with indicate the order of the model should at least eight; however, the other three criteria suggest a use were calculated using the PROC MATRIX procedure of SAS to operate program presented in Appendix Table Tests to Select the Order of the VAR Model* SMLR SMLR1 4.92 4.59 4.32 10.54 11.02 11.45 11.94 12.37 12.72 13.05 13.52 33.34 23.78 33.52 39.66 41.14 25.01 47.49 54.95 46.21 56.74 68.98 73.07 47.75 28.58 19.68 26.73 30.45 30.36 17.71 32.22 35.65 28.61 33.43 38.59 38.71 23.87 23.78 57.30 96.95 138.09 163.10 210.58 265.53 311.74 368.47 437.45 510.52 558.27 19.68 45.70 74.45 101.92 115.53 142.90 172.28 192.98 217.14 244.76 270.45 279.13 37.7 67.5 96.2 124.3 152.1 179.6 206.9 234.0 261.0 287.9 314.7 341.4 * The column heading symbols L=Number of lags, LDS=the covariance criterion, likelihood model c freedom square values matrix, LR=the ratio order relating value at indicate have SC=Schwartz' likelihood criterion, the following meanings E the determinant of s SC ratio criterion, criterion, LR1 and SMLR1=refer 2 and sequentially to the LR1 and SMLR1 percent circumstances level in which high tests of r AIC=Akaike' SMLR=Sim's to tests orders, s, X2(.05) confidence. the null error s AIC modified between a DF=degree of is the Chi Underlined hypothesis can rejected. Sims suggests that some cases advisable inves tigate significance large lags endogenous vector include those lags with significant parameters while dropping inter mediate lagged variables with insignificant parameters. s pos sibility can be considered using the LR and SMLR criteria the data in column LDS of Table 6.1. Columns LR1 and SMLR1 give the values the LR and SMLR criteria when a model of order two is tested against between a model of order two and a model of order three. Similarly, the value in column on row eight.relates a test between models order two and eight. Underlined values indicate circumstances in which the hypothesis zero values added parameters can rej ected level, i.e., circumstances in which test suggests that the model choice the higher order model. Thus the LR criteria again argues for an higher order model; however, the implications of the other three criteria an order are in higher agreement than that one. the model 1 Thus should appears not be specified inappropriate with to follow Sims s suggestion include "high"' orders lagged dependent vector present case. fact weight evidence indicates model should first order. However, con sideration of Nickelburg' finding that "underfitting of models quite common" .183) the present model is specified as a second order VAR model. Testing the VAR Model After reestimated the order of the VAR in a restricted form model that been reflects selected theoretical the model judgement relating the exogeneity of the variables in the endogenous system. In the present case this relates to the yen/$ variable. It is unlikely that the other variables in the system, with the exception trend exchange and rate intercept since variables shrimp have market significant cannot impact considered on the a major Incorporating restrictions such into nonsample estimation information mentioned parameters model necessitates use is the estimator of the restricted denoted here least that i squares s obtained estimator. when This sum squared residuals model. are minimized subject restrictions. shown various texts, including Judge 472473), the generalized least squares restricted (GLSR) estimator the solution to the minimization problem minimize over (yZ3) I)(yZfi) subject to Rfl=r, where vectorization ordinary least squares (OLS) parameter matrix, the vectorization of the OLS Y matrix, is the block diagonal matrix with matrices predetermined variables associated with each endogenous variable along the main diagonal, I is the OLS error covarinace matrix, and R is a matrix zeros ones appropriately placed to pick out the individual elements of /3 subject restrictions vector this case, r is a column vector technique zeros. of forming minimization the Lagrangian problem solving solved the partial through derivatives of the Lagrangian for 0. The result = B + (z'(lI)Z) (EQIl)z)1R l(rRf) The approach used here to first estimate fOLS and then calcu late GLSR expression shown above. ordinary least squares estimator /OLS '(E = (X 'X)X'Y, variables), the expenditures in eating places variable, the first second endogenous vector. endogenous vector composed storage of five holdings variables landings yen/dollar , imports exchange wholesale ratio. price, cold resulting estimator, fOLS, a (24 x 5) matrix parameter estimates. getting restrictions that only intercept trend variable have a significant influence on the yen/dollar ratio implies twenty exclusion restrictions parameters equation explaining fifth the yen/dollar column ratio. These estimator. parameters Following are found Bewley these restrictions can be represented '(fOLS)R2 = G, where a (20 x 24) matrix composed of null identity matrices as follows Z(8x14) Z(12x2) I(12) I(4) Z(4x5) I(4) Z(4x6) Z(4x1) Z(12x where Z(ixj) implies a null matrix with dimensions, I(4) an identity matrix of dimension x 4). a (5 x 1) vector such that R2 ' and G is a (20 x 1) null matrix. Note that since GLSR estimator employs vectorization timator restrictions must transformed erate on a column vector rather than multicolumn matrix. Bewley demonstrates this process showing that stacking the columns of RI '(fOLS)R2 yields the restrictions employed in the GLSR estimator, i.e., where es = [0o = r, Since these restrictions operate only on the parameters yen/dollar ratio equation they are known "within equation" restric tions. Alternatively, in the terminology used Bewley 82) , they "uniform mixed linear constraints" (UMLC). This convenient since Bewley 102103) shows that under special case that restrictions placed on a model are UMLC, the generalized least square restricted estimator equivalent to the maximum likelihood estimator, which implies that the restrictions placed on the model the UMLC be tested with the maximum likelihood test. Recall that the log of the likelihood function for a multivariate, normally distributed variable = n/21og(2ir)  n/21ogI  l/2(yZf)'nl(yZ/3), where a block diagonal matrix with matrices predetermined variables along its main diagonal, is the vectorization of the matrix of dependent variables is the determinant of the error covariance matrix number observations. Since term, Zp)'n1(yZfi), a scalar likelihood function pressed as L  n/21og[l where c is a constant. Thus , maximizing likelihood an observed sample equivalent to minimizing determinant of the error covariance matrix of the system. Testing the veracity of the restrictions amounts to comparing maximum likelihood values two systems: one with restrictions without. two values are significantly different, parameters of the restricted variables are assumed to be significantly are can can ex one model divided likelihood value the unrestricted model. Symbolically, the likelihood ratio = L(G*)/L(3). times natural this ratio distributed with degrees of freedom equal the number of restrictions placed on the restricted model. Thus, 21nA = 21n[L(p*)/L(/3) cn/ inln* (cn/21njn) = n [lnl0*  lnill], where the symbols remain as described immediately above. restrictions placed VAR, viz, that neither price, landings, imports, cold storage holdings , expenditures eating places, nor are assumed the monthly to be dummy correct. variables affect calculated the yen/dollar likelihood ratio ratio, value of 26.06 is not large enough to reject the hypothesis that parame ter values these variables, yen/dollar equation, are zero given twenty grees of freedom. Once restricted form model been estimated, residual vector Hoskings tested portmanteau as being statistic essentially discussed a random Chapt< process er III. using The portmanteau statistic is distributed as chisquare with k2(sp) degrees of freedom, where dimension of the endogenous vector, s was selected to be order process. present case, statistic was calculated 466.36 with grees of freedom Thus , the null hypothesis that error process Analvzine the VAR Model Table 6.2 displays parameter estimates of the restricted form of the vector autoregressive model. The tvalues give some indication proper first while, order of each this endogenous model. endogenous typically variable. variable case The tvalues Note that a high with associated with in all associated cases, tvalue; second each the first second lags each endogenous variable are underlined table. expected, variables most are parameters insignificantly different other from lagged zero. endogenous noteworthy to note that those trend parameters with high tvalues have signs that might be expected. trend imports has been positive while trend in price cold storage have been slightly negative. Concern import v over variables imports impact of imports in the price equation. Note equati that on price on and o since focuses in the price tvalues attention variables on these variables nificant. are low, However, results these since frequency relationships are relationships domain analysis not statistically appear discussed sig support below, some value to mention following points. Both sets parameters indicate a negative relationship between first order predetermined variable dependent variable. Both sets parameters also indicate a positive relationship between second order of the predetermined variable the dependent variable. 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N (0D rNO r' 4 4 *4 iU, Or c '*0 0)4 (0 *o Nt 0)o 0D N 4 *4 O r0 C N 4 IA 0 I I I F0 )0 4 D 0. . 0 0 0 I I I I I I 0n r4 C tOn oc vPt p N w in N N (00 0 0 VI 4! MN r* r 4 4. < *4 Q r (0 o . N r. N 0I O *4 ( M) 0U fO 3 i a0 a S S S in o p ra P o 0 OI CM 0 C t P 0 1 5 t I S t 1 I f t t t I SC0 N IN S0 0 0ON 000000 I St .n q O *  r V *W V l I UJ F UJ UU I. I 0 0 S00 * * 00 a . * II Ei f importer decided on a particular level import purchases. month later, when the changed level of imports begin arriving, it has effect prices. one instance, results have captured demand response the other instance, a supply response. Thus, one might conclude that within the period one month imports have inverse relationship with price but, over a two month period, imports have a positive relationship with price. final note relating stability of the VAR model is of value. Since explosive time series are atypical economic perience, it is generally believed that economic processes are stable. Thus an acceptable model an economic process should also exhibit stability. present, case, i.e., context of the model presented that in equation roots 6.1; necessary eigenvalues conditions parameter for stability matrix are lagged endogenous variables are less than one in absolute value. As shown following section, the estimated model meets these necessary conditions Frequency for stability. Domain Analysis Following process outlined in Chapter the large A matrix equation was decomposed determine frequencydetermined relationships autoregressive among the variables system. The actual in the dependent calculations vector were of the carried vector out using Gauss program detailed Appendix Table presents results of these calculations. Three frequencies were revealed, having ex degree of confidence one might place on these estimates. Accordingly, appears that the 2 month cycle the three year cycle have some validity. basis this information one might then consider the lead/lag relationships associated with each of these three frequencies. Table Frequency Relationships of the VAR Cycle Lengths tValues Phaseshift 2.31 7.34 Months 36.54 56.59 Values Landings Imports Price Storage 17.69 4.89 15.83 13.09 14.65 LeadLag Relationships Associated with the 2 .31 Month Cycle* Imports Landings +0.67 1.64 Months Imports Price Price Storage LeadLag Relationships +2.23 0.08 +1.62 0.69 Associated with +1.56 0.75 +0.95 +1.70 1.36 0.61 the 36.54 Month Cycle* Months Imports Landings +12.07 Imports Price Price 24.47 +13.68 22.86 34.92 . Storage +23.74 * 12.80 24.86 26.48 *Column variables *Underlining Lead indicates (+) or Lag () Row Variables the leadlag relationship of choice. that two possibilities are given for each pair of variables: either leads or B leads For example, the figures on rows nine ten of table indicate that landings either lead imports .67 months imports 1.64 months. Since the analysis only captures a segment of continuous time , it is impossible leading. to determine This from impossibility the phaseshift value easily which understood variable picturing segment a graph two sine waves. If the waves are out of phase, cannot determine, from information contained graph, which of the waves leading. only information contained in the graph and in the frequency analy sis presented here is the time distance phaseshift distance separating cycles two variables. Recall that absence additional information concerning the relationship between two variables, the designation of the leading variable among a given pair arbitrarily set by the convention selecting variable with the highest positive phaseshift value leading variable. However, designated lead/lag relationship between two variables can reversed subtracting the phaseshift time separating two series from a complete cycle. Given studied, some may knowled that process designation generating leading data variable being need not be arbitrary. For example, given time between ordering shipment shrimp from a foreign seller importation shrimp , it is unlikely that price is leading imports less than one one Table 6.3. It is interesting to note that this shortterm relationship between wholesale price imports appears supported information from threeyear cycle on row of Table 6.3. Note also, from data captured in the threeyear cycle rows 18 and plausibility asserting that imports lead price in the shortterm and price Table leads 6.3, imports price leads the medium imports term. indicated mediumterm on row approximately three years. A consideration of the data on the relationship between price storage argues for concluding that price leads storage approximately two months. return product. Thus, to storage shortterm operations storage is related fluctuations to the price would not expected affect price, rather, that shortterm price fluctuations would affect storage levels. Similar considerations indicate that landings storage lead leads wholesale imports price 1.35 months months. in the shortterm and meanings that most other cases leadlag in which relationship a reasonable data are somewhat argument can more be made obscure. select In those either relational possibilities, relationship choice indicated Table underlining appropriate value rows nine through twentyone. Time Domain Analysis Following relationships process among discussed the variables Chapter in the endogenous III, vector leadlag of the VAR demonstrated in Chapter III, the autocorrelation matrix relating to the vector of variables in the VAR model (6.2) = Ar(V + AVA + A2VA2' .. + AtrlVA'tr1) where r is the lag or time distance between two variables, Vts=E(UtUs) and A the parameter matrix of the VAR. autocovariance matrices relating covariances between error terms separated various lengths of time are obtained through equation this sequentially, raising value arranging r and the resulting recalculating. matrices doing in increasing order, changes one can observe as one variable covariance is compared between to larger lags two times of another series variable. The covariances or crosscovariances between two variables reveal lead/lag variables covariance relationships, (Chow, between averaged 52). over example two variables all frequencies, value at r=2, suggestion be highest between cross g that Yjt2 are most highly related, or that Yjt2 leads time periods Note that these time dimensional relationships are of general domain nature analysis while relationships are specific discovered to the identified frequency frequencies. Thus, frequency denominated relationships be expected to be more accurate than the time denominated relationships. The correlation crosscovariance between square two variables root is obtained the product dividing of the variances two series. The variances of each series are found along the main two Appendix D presents sequence autocovariance matrices time series variables endogenous system model under study here. Table .4 provides a summary the information Appendix D. Table 6.4 Autocorrelations in the VAR Model Maximum Correlation Value (r) Between Column Variable and Row Variable, Associated tValue (t) and Lag (L) at which the Maximum Correlation Occurs Landings Imports Price Storage Yen/Dollar Rate Landings Imports Price Storage Yen/S Rate 0.11 0.14 0.27 0.35 0.02 1.37 1.86 2.91 0.12 0.30 1.00 2.66 10 0.28 2.03 0.25 1.51 0 0.20 1.36 0.29 2.42 0 0.21 1.50 5 0.56 0.43 3.04 0.04 0.23 0.03 2.15 0.41 1.91 2.48 0.33 1.71 0.10 0.64 a autocorrelation b continuously in a single creasing variable crosscorrelation general , the results time domain analysis provides support support for the broadly defined relationships among variables estimated often not frequency in agreement analysis; regarding specific however details two analyses are of the relationships. For example, the data frequency in Table analysis. 6.4 indicate However, that time imports domain lead price analysis indi cates that imports lead price ten months rather than the 1. 55 months indicated by the frequency analysis. similarly, both analyses indicate that landings lead wholesale price; but, time domain analysis indicates the timedistance separating two variables is five months rather than the 2.2 months suggested the frequency domain analysis. indicates that, mediumterm, imports lead storage approximately twelve months. The time dimension analysis estimates time distance between these two variables to be five months. some cases the results of the two types of analysis appear agreement. example , both time frequency domain analysis indicate that price leads storage approximately nine or ten months. results two analyses are also reasonably close concerning frequency relationship analysis between indicates the yen/dollar a leadership ratio role and imports. the yen/dollar ratio approximately seven months. time domain analysis indi cates time distance between two variables to be four months; however, crosscorrelation function between two variables sharply peaked four month lag. fact, difference between the crosscorrelation value at the four month interval and the seven month interval amounts to only four percent correlation value at the four month interval (see Appendix In those incidence where results time domain analysis frequency :requenc based domain results. differ, a knowledge For example, in the of the market case favors of the relationship between imports wholesale price, frequency analysis result, that imports lead price 1.55 months, appears more plausible than the alternative that imports lead price ten months. Given that supplies shrimp are regularly added storage during spring, summer, fall harvest seasons during winter not new 