An Evaluation of Econometric Models of U.S. Farmland Prices Rulon D. Pope, Randall A. Kramer, Richard D. Green, and B. Delworth Gardner Previously published empirical models of U.S. farmland prices are reviewed and reestimated including recent data. It is apparent that structural changes have occurred. A simple single equation econometric model with less economic structure appears to forecast better than a simultaneous equation model. Finally, Box-Jenkins forecasts are roughly as good as those based upon a simultaneous equation econometric model, but somewhat inferior to the single equation model. The results suggest that further research may be needed to explain recent movements of farmland prices. During the post World War II era, there has been much concern over rapidly rising agricultural land prices. Research by agricul- tural economists on the determinants of farmland prices appears to have peaked dur- ing the 1960's [Brake and Melichar]. How- ever, the recent escalation in the rate of farmland price increases has caused a re- surgence of interest [Morris, Harris, Gard- ner]. Though concern has been expressed over the ability of farm income to support farm- land prices, there has been little effort de- voted to the evaluation of econometric models of these prices in light of recent ex- periences. Clearly, policy decisions regard- ing commodity programs, financial instru- ments, and the impact of macroeconomic forces on farmland values and the distribu- tion of wealth would benefit from a clear understanding of the farmland market. The concern of this paper is whether previously The authors are Assistant Professor, Post Graduate Re- search Associate, Assistant Professor and Professor, re- spectively, Department of Agricultural Economics, University of California, Davis. Giannini Foundation Research Paper No. 524 The authors gratefully acknowledge helpful comments by the Editor and anonymous reviewers. published models of the farmland market re- tain their structural credibility when esti- mated with recent data. The approach fol- lowed is not to critically evaluate or revamp these earlier models, but to examine their plausibility as explanations of recent market events, and study their predictive ability. After analyzing the reestimated models, one simultaneous equation and one single equa- tion model are used in the evaluation of fore- casting performance. To facilitate the analysis, a naive forecasting model (Box- Jenkins) without economic structure is com- pared with the econometric models. The naive forecasts are considered as benchmark results when evaluating the performance of the econometric models. Some Econometric Models of Farmland Prices Several simultaneous equation models of the U.S. farm real estate market have been developed. Three of the best known models are those presented by Reynolds and Tim- mons, Tweeten and Martin, and Herdt and Cochrane. All of the models did a reasonable job of explaining variations in land prices dur- ing the period for which they were originally estimated. To determine how well the models might perform now, they were rees- timated utilizing more recent data. These 107
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An Evaluation of Econometric Models ofU.S. Farmland Prices
Rulon D. Pope, Randall A. Kramer,Richard D. Green, and B. Delworth Gardner
Previously published empirical models of U.S. farmland prices are reviewed andreestimated including recent data. It is apparent that structural changes have occurred.A simple single equation econometric model with less economic structure appears toforecast better than a simultaneous equation model. Finally, Box-Jenkins forecasts areroughly as good as those based upon a simultaneous equation econometric model, butsomewhat inferior to the single equation model. The results suggest that further researchmay be needed to explain recent movements of farmland prices.
During the post World War II era, therehas been much concern over rapidly risingagricultural land prices. Research by agricul-tural economists on the determinants offarmland prices appears to have peaked dur-ing the 1960's [Brake and Melichar]. How-ever, the recent escalation in the rate offarmland price increases has caused a re-surgence of interest [Morris, Harris, Gard-ner].
Though concern has been expressed overthe ability of farm income to support farm-land prices, there has been little effort de-voted to the evaluation of econometricmodels of these prices in light of recent ex-periences. Clearly, policy decisions regard-ing commodity programs, financial instru-ments, and the impact of macroeconomicforces on farmland values and the distribu-tion of wealth would benefit from a clearunderstanding of the farmland market. Theconcern of this paper is whether previously
The authors are Assistant Professor, Post Graduate Re-search Associate, Assistant Professor and Professor, re-spectively, Department of Agricultural Economics,University of California, Davis.
Giannini Foundation Research Paper No. 524
The authors gratefully acknowledge helpful commentsby the Editor and anonymous reviewers.
published models of the farmland market re-tain their structural credibility when esti-mated with recent data. The approach fol-lowed is not to critically evaluate or revampthese earlier models, but to examine theirplausibility as explanations of recent marketevents, and study their predictive ability.After analyzing the reestimated models, onesimultaneous equation and one single equa-tion model are used in the evaluation of fore-casting performance. To facilitate theanalysis, a naive forecasting model (Box-Jenkins) without economic structure is com-pared with the econometric models. Thenaive forecasts are considered as benchmarkresults when evaluating the performance ofthe econometric models.
Some Econometric Models ofFarmland Prices
Several simultaneous equation models ofthe U.S. farm real estate market have beendeveloped. Three of the best known modelsare those presented by Reynolds and Tim-mons, Tweeten and Martin, and Herdt andCochrane. All of the models did a reasonablejob of explaining variations in land prices dur-ing the period for which they were originallyestimated. To determine how well themodels might perform now, they were rees-timated utilizing more recent data. These
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models are briefly reviewed and the resultsof the reestimation are discussed below.
Reynolds and Timmons used a two-equation recursive model for identifying theprincipal determinants of agricultural landprices for the period 1933-1965. They foundthat much of the variation in land pricescould be explained by expected capital gains,predicted voluntary transfers of farmland,government payments for land diversion,conservation payments, farm enlargement,and the rate of return on common stock.When the model was reestimated with morerecent data, 1946-1972, there were a numberof changes in the signs and magnitudes of thecoefficients. Table Al in the Appendix detailsthe definitions of variables used and the re-gression results. In the price equation four ofthe eight signs reversed and only one coeffi-cient is statistically different from zero at thefive percent level.
Tweeten and Martin presented a five-equation model for explaining changes infarmland values over time using recursiveand ordinary least squares.1 They found thatthe two major determinants of farm real es-tate price increases between 1950 and 1963were capitalized benefits from governmentprograms tied to land and pressures for farmenlargement. The model has been reesti-mated for the more recent period (1946-1972). Variable definitions and presentationof these regression results are found in TableA2 in the Appendix. Again, there was anabundance of sign changes and lack of statis-tical significance. For example, regardless ofthe estimation technique, all coefficients ex-cept lagged price (and possibly farm num-bers) are statistically insignificant in the priceequation.
The final simultaneous equation modelconsidered is one presented by Herdt andCochrane. They concluded that technologicalprogress in conjunction with government
Tweeten and Martin also employed a correction for au-tocorrelation for ordinary least squares. However, theydiscarded the model with autocorrelation when examin-ing forecasting performance.
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supported output prices led to rising farm-land prices. As with earlier models, thismodel also encounters problems when esti-mated with more recent data. However,generally sign reversals were few in numberand lagged dependent variables are not used.For these reasons, this model was chosen as arepresentative of the simultaneous equationapproach to be analyzed in more detail andutilized in forecasting later on. 2 Model formu-lation and estimation results are discussed ingreater detail below.
The model is defined by
Ns = f(P,R,U,Lf)(supply equation)
Nd = f(P,R,T,Pr/Pp,G)(demand equation)
Ns = Nd(identity)
where Ns is the number of farms supplied;3
Nd is the number of farms demanded; P isthe average value per acre of U.S. agricul-tural real estate in current dollars; R is therate of return on nonfarm investment; U isthe unemployment rate; Lf is the amount ofland in farms; T is the USDA productivityindex; Pr/Pp is the ratio of the index of pricesreceived by farmers to the index of pricespaid by farmers; and G is the wholesale priceindex.
Herdt and Cochrane estimated this modelfor 1913-1962 using two-stage least squares(2SLS). The model has been reestimated forthe post-war years 1946-1972. In addition,the model has been estimated for 1913-1972.Since, a priori there is no reason to assumean absence of correlation across equations,both 2SLS and three-stage least squares(3SLS) estimates for the two time periods, aswell as the Herdt and Cochrane original es-timates, are presented in Table 1.
2A major reason for distinguishing the Herdt and Coch-rane model from the other simultaneous equationmodels is its dissimilarity to Klinefelter's single equa-tion model to be discussed later.
3Due to data limitations, farmland sold can only becharacterized by the number of transfers, rather thanacreage sold.
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There is one sign change in each of thefour sets of new estimates. In two of the setsof estimates, the sign of the coefficient fornumber of transfers in the demand equationreversed. This sign change may not be par-ticularly meaningful since these estimates,and the original estimate of Herdt and Coch-rane, are not significantly different from zeroat conventional levels of type I errors. Also,the sign of the coefficient for the wholesaleprice index changed. Of the four new esti-mates for this coefficient, two are positive,two are negative, and all are statistically in-significant. Hence, the addition of ten moreyears of data makes it difficult to argue thatthis coefficient is nonzero.
In addition to these econometric models, arecent single equation model was testedwhich has less structural content than theother models, but fits the data well. Klinefel-ter assumes that the number of farm transfersis exogenous. Although the model is quitesimple, the results have generated profes-sional interest [Brake and Melichar].Klinefelter found that 97 percent of the varia-tion in Illinois land prices between 1951 and1970 could be explained by net returns, aver-age farm size, number of transfers, and ex-pected capital gains. A model similar toKlinefelter's was estimated for U.S. data forthe periods 1946-1972 and 1913-1972.4 Theresults are presented in Table 2.
For the 1913-1972 estimates, the coeffi-cients for net farm income and average farmsize have unexpected signs. The coefficientsfor average farm size and number of transfersare not significantly different from zero.When the model was estimated for 1946-1972, the expected signs for net farm incomeand average farm size were obtained, but thesign for number of transfers reversed. De-
4The model differed from Klinefelter's model as follows:(a) net farm income was used in place of net returns tolandlords; (b) instead of deflating variables, the GNPdeflator was entered as an explicit variable; (c) in thecalculation of capital gains, capital improvements weresubtracted out. These changes were required either toaccommodate U. S. data or to achieve consistency withthe other model used in forecasting.
spite several implausible signs, this modelwas utilized for forecasting due to the highpercentage of price variation explained bythe variables (R2 of .952 (1913-1972) and .989(1946-1972)) and for the reasons mentionedabove. 5
Forecast Results- Econometric Models
In order to forecast with the Herdt andCochrane model, the reduced form equationfor price was calculated and then the valuesof the exogenous variables were substitutedto solve for price. Thus, the forecasts are expost in the sense that actual values of theexogenous variables are used. The results arepresented in Table 3. On the basis of rootmean square error (RMSE), the various ver-sions of the Herdt and Cochrane model canbe compared. It is apparent that for within-sample forecasting, both sets of 2SLS esti-mates outperformed the 3SLS estimates.This is a rather surprising result sinceeconometricians generally prefer 3SLS over2SLS due to a presumption of the latter's lackof asymptotic efficiency. However, the bet-ter forecasting performance of the 2SLS es-timates may result from the fact that full in-formation estimation methods, such as 3SLS,are more sensitive to specification error thanare the k-class estimators such as 2SLS. Since3SLS takes into account the correlation be-tween the disturbances of all the structuralequations, a specification error in one equa-tion will affect all of the coefficient estimatesof the system.
None of the Herdt and Cochrane reducedform equations forecasted well beyond thesample. For example, the actual undeflatedvalue of farm real estate per acre was $340.48in 1975. The highest forecast for that year was
5 0f course a high R2 does not necessarily imply that amodel will forecast well. Moreover, the Durbin-Watsonstatistic for the period 1946-1972 suggests evidence ofpositive autocorrelation. However, since expected capi-tal gains contains transformations of lagged values ofthe dependent variable, the Durbin-Watson statisticmay not be appropriate. If autocorrelation is present,parameters estimates will be inconsistent.
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TABLE 1. Estimation Results for the Herdt and Cochrane Modela
aForecasts of undeflated value of U.S. agricultural land and buildings per acre.bRoot mean square error.
$269.40, and the lowest was $222.07. On thebasis of RMSE beyond the sample, the 2SLS1946-1972 estimates performed the best, fol-lowed by the 3SLS, 1946-1972 estimates.
The forecast results for the modifiedKlinefelter model are also presented in Table3. For within-sample forecasts, the 1946-1972 estimates did better than any of theHerdt and Cochrane reduced forms. Forboth time periods the Klinefelter modelforecasted better beyond the sample thaneach of the Herdt and Cochrane versions. Itis apparent from these results that a simplemodel with implausible signs (e.g., coeffi-cient of transfers) can still forecast quite well.
In the modified Klinefelter model ex-pected capital gains (a three year moving av-erage) includes lagged values of the depen-dent variable price. One is curious whethertime series models based solely on the lagstructure of the dependent variable plusmore general error structures might possessas great a predictive power as the economicmodels. In the following section, Box-Jenkinsforecasts are presented and later comparedwith the econometric forecasts. The Box-Jenkin's results are viewed as benchmarkforecasts, since it is generally hoped thateconometric models perform at least as wellas naive statistical models.
Forecast Results - Time Series Model
As an alternative to the econometric
models, time series models of an integratedautoregressive moving average form are usedto obtain forecasts of land prices [Box andJenkins]. 6 These are statistical models of theform
where the Z's are observations generated bya stochastic process, the U's are indepen-dently distributed random variables withmean zero and constant variance, and 8, 0i,and Oi are unknown parameters.
The first part of the model is referred to asthe autoregressive portion and the latter partas the moving average portion. The term au-toregressive derives from the fact that thefirst part of the above model is essentially aregression equation in which Zt is related toits own past values rather than a set of inde-pendent variables. The moving average termcomes from the fact that the second portionof the time series model is just a moving av-erage of the disturbances reaching back for qperiods. If the observations are in differenceform, then the process is called an integratedautoregressive-moving average process
6 Agricultural economists are not as familiar with thistechnique as with econometric models; however, therehave been some applications of this procedure in ag-ricultural economics. See, for example, Oliveira andRausser, and Schmitz and Watts.
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(ARIMA). Differencing of the data is oftennecessary in order to convert the process intoa stationary one; that is, to yield a time seriesin which the joint distribution of any subsetof observations of the series remains un-changed when the same constant is added tothe time subscript of each observation. A re-lated concept is that of weakly stationary inwhich a time series mean and autocovariancefunction are independent of time. In mostcases first or second differencing of the origi-nal series suffices to convert a nonstationarytime series into a stationary one. From anestimation viewpoint, stationarity results in areduction of the number of unknown param-eters to be estimated.
The first stage in selecting an appropriatetime series model is to properly identify theprocess generating the observations. This isdone by examining the estimated autocorre-lation and partial autocorrelation functions.Box and Jenkins (pp. 176-77) provide tablesdescribing the nature of the theoretical au-tocorrelation functions for various ARIMAprocesses. In general, the identification pro-cedure is based on the characteristic behaviorof autocorrelations and partial autocorrela-tions for known ARIMA processes. Forexample, if the autocorrelations exhibitspikes at lags 1 through q, then cut off andthe partial autocorrelations tail off, then theprocess is a qth order moving average pro-cess. Other characteristics describe auto-regressive and mixed autoregressive-movingaverage processes. These characteristics,however, describe the behavior of theoreti-cal autocorrelation and partial autocorrelationfunctions. To analyze the estimatedfunctions, the standard errors are needed todetermine if the spikes are significantly dif-ferent from zero at various lags.
From the estimated autocorrelation andpartial autocorrelation functions, given inTable 4, based on 1913-1972 observations,the model was identified as an ARI(2,2) orpossibly an IMA(2,2); that is, an integratedautoregressive process of order 2,2 or an in-tegrated moving average process of order2,2. To see this, observe the patterns of the
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autocorrelation and partial autocorrelationfunctions in Table 4. For the original seriesthe autocorrelations remain relatively largefor several lags suggesting a nonstationary se-ries. The standard errors associated with thefirst and second lags are 0.13 and 0.21 re-spectively. Thus, at least one differencing isnecessary. For the first difference, only theautocorrelations of lags three and five aresignificantly different from zero. Their as-sociated standard errors are 0.13 and 0.14.This may suggest some first order process,but based on graphical inspection of the firstdifference and that of the second differencetogether with the autocorrelations and partialautocorrelations, a higher order model ap-peared to be more appropriate.
The autocorrelations of the second differ-ence are relatively large for the first two lagswith standard errors of 0.13 and 0.14 respec-tively. However, there are some rather largevalues of the autocorrelations for larger lagssuggesting a mixture of exponentials ordamped sine waves. The same conclusionsare reached by considering the values of thepartial autocorrelations. Thus, these observa-tions suggest an ARI(2,2) or possibly anIMA(2,2) process.
Based on these identifications, the modelswere estimated. The estimated results are:
Wt = 1.071 Wt-1 - 0.215 Wt_2 + UtARI(2,2) model,
Wt = Ut + 1.104 Ut-1 - 0.023 Ut-2IMA (2,2) model,
where Wt represents the second difference ofthe Zt.
Diagnostic checks were made on both ofthe estimated models. The estimated au-tocorrelations of the residuals were used toevaluate the goodness of fits of the models.If, for example, Wt = 1.071 Wt-1 - 0.215Wt_2 + Ut were approximately equal to thetrue model, then the estimated residuals, 0t,would constitute a white noise process. Thus,the sample autocorrelations of the estimatedresiduals would be approximately uncorre-lated. Any departure from small values of au-
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Western Journal of Agricultural Economics
tocorrelations would indicate inadequacies inthe fitted model. In both models, all of theautocorrelations were not significantly differ-ent from zero. In addition, the Box-Pierce Qstatistic
KQ =T r r2
j=l
where rj are the estimated autocorrelations ofthe residuals, T = number of W's used to fitthe model, and Q is approximately chi-square distributed with (K-p-q) degrees offreedom, was calculated to determine thegoodness of fit of the models. 7 Using K = 20,the number of lags, for both models the valueof Q, 17.75 and 13.14, was small relative tothe critical value of X205,18 = 28.9. The valuesof the Box-Pierce Q statistic indicate that thewhole set of sample autocorrelations for lags 1through 20 taken as a whole are small. Thus,the residuals from both of the models, basedon the Q statistic, tend to indicate that themodels fit the observed data well.
7 For a more detailed discussion of the use of the Qstatistic see Box and Jenkins, pp. 290-291.
The forecasting performances of the aboveestimated models were examined by predict-ing land prices within and outside the sam-ple. The within sample forecasts were basedon one period forecasts for the period 1913-1972, and the forecasts outside the samplewere for the years 1973, 1974 and 1975. Theresults are presented in Table 5. For com-parative purposes, results from a logarithmicmodel for the years 1913-1972 are also pre-sented. Though substantially reducing the de-grees of freedom, the postwar years werealso estimated separately because of the landprice spiral during this period.
The empirical results show that all of themodels performed much better within ratherthan outside of the sampling period. How-ever, the outside forecasts are one, two, andthree period ahead forecasts, whereas thewithin sample forecasts are all one periodahead forecasts. Furthermore, forecasts ob-tained from the estimated model based on1946-1972 data appear superior to those ofother models. However, the estimatedstandard errors of the coefficients were highdue to the relatively small number of obser-vations used. The rule of thumb in estimat-
aRoot mean square error.bupper and lower limits for 95 percent confidence intervals.CModel based on data in logarithmic form.dModel based on 1946-1972 sample period.
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Pope, Kramer, Green, and Gardner
ing time series models is that at least 50 ob-servations are needed to adequately estimatea model [Box and Jenkins, p. 18].
The logarithmic model performs relativelywell. On the basis of RMSE, it outperformsall other time series models including thepostwar model. The implication of thelogarithmic model is that percentage changes(rather than the level of changes) have re-mained relatively stable through time.
A Brief Comparison of Econometric andTime Series Forecasts
The simultaneous equation econometricmodel used in this study yielded forecastsabout as accurate as the benchmark forecastsof the Box-Jenkins method for the postwaryears when land prices were rapidly escalat-ing. For the longer time period (1913-1972),the time series models performed better thanthe simultaneous equation model on the basisof RMSE, both within and beyond sample.For this same period the Klinefelter modelhad the lowest beyond-sample RMSE. Fur-ther, the Klinefelter model performed betterthan either time series or the simultaneousequation econometric model for the postwaryears. Overall, the poorest predictors appearto be generated by the simultaneous equa-tion models.
Conclusions
It is not uncommon when comparing timeseries and econometric forecasts to discoverthat time series models provide as good orbetter short-term forecasts than econometricmodels.8 The above results are consistentwith this conclusion. However, the singleequation model predicted well too, and itmay generate the best predictors. This resultis surprising, particularly since the mag-nitudes and signs of the coefficient estimatesappear very sensitive to the sample periodused. Also, although the model may have mi-
8 For an interesting discussion of the relative merits ofBox-Jenkins versus econometric models see Naylor,Seaks, and Wichern.
croeconomic foundations, it explains little asa market model. Since expected capital gainsare functionally related to lagged land prices,it appears that more study is needed to ex-plain the recent rise in farm prices and capitalgains.
The simultaneous equation models pre-sumably have greater causal foundations re-flecting market behavior of sellers andbuyers. However, results in Tables 1, Al andA2, indicate that attempts to incorporategreater structural detail in econometricmodels of the land market have not instilledmuch confidence in their structural perform-ance. One would expect the magnitude of pa-rameter estimates to be sensitive to the sam-ple period. However, when recent data wereadded to the sample numerous changes insigns of coefficients occurred for all of thesimultaneous equation models. Further,most of the estimated coefficients were notstatistically significant from zero.
These results suggest that the model speci-fications do not reflect accurately enough therelevant structural changes and other charac-teristics of the farmland market. Therefore, ifone is concerned with both predictive abilityand economic structure, additional researchis needed to explain recent movements offarmland prices. Such research may seek abetter understanding of how expectations areformulated by land market participants, and abetter understanding of the motives for hold-ing real versus liquid capital in an inflationaryeconomy. Further effort directed towards aset of more general statistical assumptionsmay also prove fruitful. For example, onecould integrate the ARIMA Box-Jenkinsmodels and econometric models as recentlysuggested by Newbold and Davies. Giventhe interest in the farmland market,additional research should be beneficial tofarmers and policy makers.
References
Box, G. E. P., and G. M. Jenkins. Time Series AnalysisForecasting and Control. San Francisco: Holden-Day,1970.
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Brake, J. R., and E. Melichar. "Agricultural Financeand Capital Markets," in L. R. Martin (ed.), A Surveyof Agricultural Economics Literature, Volume 1, Uni-versity of Minnesota Press, Minneapolis, 1977.
Gardner, B. D. "Issues Affecting the Availability andPrice of Land for Agriculture." Paper presented,California Chapter Meeting of the American Societyof Agronomy, Fresno, January 26, 1978.
Harris, D. G. "Inflation-Indexed Price Supports andLand Values." American Journal of Agricultural Eco-nomics, 59 (1977):489-495.
Herdt, R. W., and W. W. Cochrane. "Farmland Pricesand Technological Advance." Journal of Farm Eco-nomics, 48 (1966):243-263.
Klinefelter, D. A. "Factors Affecting Farmland Valuesin Illinois." Illinois Agricultural Economics, 13(1973):27-33.
Morris, D. "Farmland Values and Urbanization." Ag-ricultural Economics Research, 30 (1978):44-47.
Naylor, T. H., T. G. Seaks, and D. W. Wichern. "Box-
Jenkins Methods: An Alternative to EconometricModels." International Statistical Review, 40(1972):123-137.
Newbold, P. and N. Davies. "Error Misspecification andSpurious Regressions." International Economic Re-view, 19 (1978):513-519.
Oliveira, R. A. and G. C. Rausser. "Daily Fluctuationsin Campground Use: An Econometric Analysis."American Journal of Agricultural Economics, 59(1977):283-293.
Reynolds, T. E., and J. F. Timmons. Factors AffectingFarmland Values in the United States. Iowa Agricul-tural Experiment Station Research Bulletin 566, 1969.
Schmitz, A. and D. G. Watts, "Forecasting WheatYields: An Application of Parametric Time SeriesModeling." American Journal of Agricultural Eco-nomics, 52 (1970):247-262.
Tweeten, L. G., and J. E. Martin. "A Methodology forPredicting U.S. Farm Real Estate Price Variation."Journal of Farm Economics, 48 (1966):378-393.
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