Forecasting an agricultural system with random walk priors

4gricultural Systems 21 (1986) 59-67

Forecasting An Agricultural System With Random Walk Priors D a v i d A. Bessler & Jane C. H o p k i n s Department of Agricultural Economics, Texas A&M University, College Station, Texas, USA

S UMMA R Y The paper explores the use of random walk priors for specifying multiple time series models. An empirical example with the US shrimp market is presented. Out-of-sample forecasting results offer encouragement for further consideration of these models.

F O R E C A S T I N G AN A G R I C U L T U R A L SYSTEM WITH R A N D O M W A L K PRIORS It has now been over fifteen years since economists began to utilize time series methods as a serious alternative to standard "structural' econometric methods. While the economics and the agricultural economics professions are by no means in unanimous agreement with regard to their use, one can hardly pick up a copy of a current journal dealing with empirical work and not see at least one paper utilizing time series methods. Perhaps one reason for their increased use is that time series methods have demonstrated success in actual forecasting competitions with structural models (see Granger & Newbold, 1977, Chapter 8, for a discussion of some early forecasting competitions). One major difference between the time series method and the standard 'structural' econometric method is that the former relies on empirical regularities for explicit model specification, while the latter uses prior economic theory for specification. Accordingly, the time series approach to economic model building is often characterized by its critics as mere 59 Agricultural Systems 0308-521X/86/$03.50 ~ Else~ier Applied Science Publishers Ltd, England, 1986. Printed in Great Britain

60

David A. Bessler, Jan~" C. Hopkins

data mining or rabbit hunting (following Learner (1976) we should note that mining and hunting are not altogether non-productive activities). Recent innovations in time series analyses utilize more prior information in model specifications. That is to say, research workers have recently started to put prior restrictions (probabilistic restrictions) on the degree of series interactions in analysis of multiple economic time series. Litterman (1979), in particular, has investigated the use of a random walk prior. This prior has shown promise in out-of-sample forecasts from macroeconomic models (Doan et al., 1984) and from a model of the US hog market (Kling & Bessler, 1985). This prior has resulted in significant improvement in out-of-sample mean squared error measurements-especially at long forecast horizons. In this paper we consider a general case of the Litterman (1979), random walk prior. Here we wish to contrast priors based on previous forecast experience (symmetric priors) with priors based on expert information (non-symmetric priors). In more technical terms, we contrast the symmetric Litterman prior with a non-symmetric prior (defined below). Both are random walk-type priors. The paper is organized into three sections, the first of which is this introduction. Next, we present the random walk prior in both its symmetric and non-symmetric forms. Following the discussion of the prior, we apply both forms to multiple time series analysis of recent data from the US shrimp market. Out-of-sample forecasts from these representations are compared with those based on unrestricted multiple time series representations and univariate time series models. The final section of the paper discusses alternative types of priors which may prove useful for further investigation.

R A N D O M WALK PRIORS The use of unrestricted classical least squares regression on sample data is equivalent to the use of Bayesian models, where the sample data is combined with a diffuse prior, having a center on zero and infinite variance (Zellner, 1971, p. 20). Accordingly, where researchers approach data with no prior information on magnitudes or signs of particular model coefficients, Bayesian methods otter little, relative to classical least squares. Even where researchers have non-diffuse, yet non-dogmatic priors.

Forecasting an agricultural system with random walk priors

6I

classical least squares estimates are limiting cases, when the sample is permitted to grow without finite bound. A non-dogmatic prior puts a positive probability density (however small) on any point interval on the real line ( - :c, + :c) for each parameter to be estimated. That is, classical least squares estimates become the limiting distribution for Bayesian methods, when the latter are performed with arbitrarily large data sets. Bessler (1980), for example, uses this latter argument to support the use of classical methods in modelling farmers" subjective probability distributions. In many cases, research workers have non-diffuse priors and a limited number of data points (relative to the number of parameters they wish to estimate). In such cases, Bayesian methods will give results which may differ from those obtained with classical least squares estimation. Litterman (1979) argues that analysis of multiple economic time series is a case where both of the above conditions hold. We usually have a limited number of data points. Say, for example, one wishes to study the US hog market in a five-variable six-lag vector autoregression with postWorld War II quarterly data (as did Bessler (1984a)). Such an effort results in (5 x 6 x 5) = 150 estimates of lagged coefficients ptus (5 x 6 + 2 ) = 15 distinct estimates of the elements of the contemporaneous covariance matrix. With post-World War II data. say up through 1981, this results in a data-to-parameter ratio of about four (a dismally low ratio, Zellner, 1982, p. 314). Fortunately, research workers on multiple economic time series have fairly strong priors on the time series properties of many of their data. In particular, many economic time series are approximately well-modelled as random walks (Pt = Pt-~ + et, where ¢t is white-noise innovation). This prior is held for two (what are surely related) reasons. First. the random walk behavior of asset prices traded in competitive auction-type markets is an implication of the efficient market hypothesis (Samuelson, 1965). And, even in cases where one would not expect the efficient market hypothesis to hold exactly (agricultural prices with significant storage costs), a random walk prior may be much better than a prior which centers coefficients of all lagged variables on zero (classical least squares). A second reason for the random walk prior relates to the empirical evidence uncovered on the univariate properties of many economic time series. Typically, economic time series exhibit near non-stationary behavior--requiring first differencing to induce stationarity lsee Granger, 1966). Bessler (1982), for example, studied 24 price and yield series and

62

Dacid ,4. Bessler, Jane C. Hopkins

argued that many required first differencing. Plosser & Schwert (1978) make a similar argument--that first differences are usually required for analysis of many economic time series. The general multivariate time series model is given in its autoregressive form in eqn (1): X, =

da(s)X,_s +

(l)

s=l

where X~ is a (k x 1) vector of variables measured at period t, 4,(s) is a (k x k) matrix of autoregressive coefficients, which relate X~ to X~_s and e, is a (k x 1) vector of white-noise disturbances. Under usual methods, eqn (t) is approximated by fitting a mth order autoregression (s = m), using classical least squares regression equation-by-equation (Bessler, 1984b. pp. 26-27). Litterman proposes a prior on the coefficients ofeqn (I) characterized as follows: The ~'s are jointly normally distributed. The mean of ~ij(t) is zero except that on ~,(1), which has a mean of one (here (b~j(t) is the i. j element of the autoregressive matrix at lag t). In addition to the above information on the center of the prior distribution, the following tightness information is specified: 2, a constant standard deviation on the coefficient associated with the first lag of the dependent variable (overall tightness). The standard deviations of all coefficients in the lag distributions are decreased in a harmonic manner, according to the parameter "/~ (decay tightness). Standard deviations on other variables in the system can be made tighter than own lag distributions according to the parameter ;'2 (interaction tightness). Given the above parameters ().. ,,~ and 72), the standard deviation of coefficient i, j at lag l will be given a s [

if: i = j Tt

6,5 =.

(21

Z"eai [;'~G;

if: i # j

Forecasting an agricultural system with random walk priors

63

where a,. and aj are the standard deviations on innovations from univariate autoregressions for equations i and j, respectively. Their purpose is to scale the prior in eqn (2) for relative size of the original series. The prior summarized above treats each series symmetrically. That is, "1'2, the tightness parameter associated with coefficients of variable i in equation j, is the same for all i and j. Often, one has prior information which suggests that the above may not be reasonable. For example, one might expect one variable to be near exogenous in a system--and thus may wish to put a very tight prior around zero on coefficients on other variables in its equation. While, on other equations, the researcher may be quite uncertain on coefficients of lagged variables--and thus may wish to impose a rather loose prior. Accordingly, eqn (2) may be modified using equation specific interaction tightness; "./2(i, j) reflects tightness information on coefficients of variable j in the i~h equation of the vector autoregression. Values of 7z(i,j) between (1,0) will reflect more (1) or less (0) series interactions. The symmetric prior--that given by tightness information in eqn (2)-has been studied by Bessler & Kling (1985) for the US hog market. They find rather mixed results. The symmetric prior outperforms an unconstrained vector autoregression in out-of-sample forecasts;while it does not outperform forecasts based on univariate autoregressions. The non-symmetric prior outperforms both the unrestricted VAR and the symmetric prior of long forecast horizons. Doan et al. (1984) find the symmetric prior does quite well in out-of-sample forecasts of macroeconomic series. Below we consider both the symmetric and nonsymmetric random walk priors in forecasts from a five-variable model of the US shrimp market.

APPLICATION TO US SHRIMP MARKET Five variables relevant to the US shrimp market were modelled in a vector autoregression. The variables are shrimp landings, imports, price, stocks and US disposable income. The data are measured monthly from January, 1964 to December, 1982. More explicit data descriptions can be found in Hopkins (1983). The series were modelled in four alternative ways--in an unrestricted vector autoregression, in a vector autoregression with a symmetric random walk prior, in a vector autoregression with a non-symmetric

64

Dacid A. Bessler, Jane C. Hopkins

random walk prior and in five univariate autoregressions. The models were fit over periods 1964-1979. Lag length (s of eqn (1)) was selected for the unrestricted VAR, using sequential likelihood ratio tests (see Bessler, 1984b). The test was run over s - I . . . . ,18. A thirteenth-order VAR was selected for further analysis. Out-of-sample forecasts were then made over the 36 periods, 1980-1982. Forecasts were made at various horizons. Results are reported at 1-, 2-, 6-, 12- and 18-month forecast horizons. All estimation and forecasting was carried out using Theil mixed estimation as programmed in Doan & Litterman (1981). As we had very little experience with the symmetric priors used in this study, an empirical 'fitting' procedure was used. That is, we fit the vector autoregressions with random walk priors over the period 1964-1976 for various settings on the parameters 2, 7t and 7_, of eqn (2). We then forecasted out-of-sample the 36 data points observed over 1977-1979. Values of parameters 2, "~'1 and ;;'2 which gave the minimum log determinant of error covariance matrix (see Doan et al., 1984 for details), were used in the forecasting competition over the final 36 observations (monthly observations over 1980-1982). The optimum values of symmetric prior parameters were found to be 2 =0.25, "~'1= 0 . 0 0 and 5'_, = 0-25. The non-symmetric prior used the same values, 2 = 0.25 and 5'1 = 0"00, found above with values of 7z(i,j) as summarized in Table 1. As one can see, income is treated as nearly exogenous, while all variables are permitted to influence price with about the same prior tightness. TABLE

1

Setting of Interactions Among Series for Non-Symmetric Prior Distribution" Lagged cariables

Landings Price Imports Stocks Income

Dependent cariables Landings

Price

Imports

Stocks

Income

1-00 0-80 0-0 l 0.01 001

0.80 1.00 0-80 0.80 0-80

0.01 0.80 1.00 0.01 0.01

0-80 0.01 0.80 I 00 0-01

0.01 0.01 0-01 0.01 1.00

a Entries in the Table reflect the relative degree of series interactions permitted using the non-symmetric prior distribution. A value of 1.0 is full interaction. Values close to zero imply little series interaction is permitted.

6"58

4.86 6-31 5.211

4"93

3'56 4.20 3.76

5"96 4.14 5.46 5.93

0.47 0-46 1).61 0.60

4.44 3-71 4-1111 3.99

2 months

7-17 12.95 8'24

14"61

6'93 4"43 5.73 6.54

/}.56 0.52 0.81 0.77

6.02 5.01 4.25 4.44

6 months

Forecasl h o r i z o ,

" Results on income are not given as we are not explicitly interested m forecasling income.

Unrestricted 13-lag VAR Symmelric l/aycsian 13-lag VAR General Bayesian 13-lag VAIl Univariate

Sloeks

Unrestricted 13-lag VAR Symmctric Baycsia|l 13-lag VAR General Bayesian 13-lag VAR Uniwmalc

5'22 3"73 4.73 5.211

0.36 0.35 1t-42 0-45

Price Unrestriclcd 13-lag VAR Symmetric Ih|yesia|| 13-lag VAR General Bayesian 13-lag VAR Uniwlriate

hnpor|s

3-81 3-33 2.99 2.96

I monlh

Landings Unrestricted 13-lag VAR Symmetric Bayesian 13-lag VAR General llayesia|| 13-lag VAR Uniwlriale

Series~type o1 mo~h'l

8-81 14.80 8.54

15-43

5'94 4.77 5-92 6.75

0.70 0.52 0.88 1.07

5-90 3.58 4.41 4.28

12 lllOlllh.¥

15.32 21.76 9.86

II "85

6'26 6.115 6.61 7.41

0.90 0-63 I.OI 1.52

5.71 5.53 5.21 5.66

1~ tl~lOlllh,s'

TABLE 2 Out-of-Sample I:orecasl Perlbrmance: Ct)mparisons of Root Mean Square Errors I\~r Alternative Time Series Models*

'::'

:~

2~

,~

"

u~

~ ~"

66

Darid .4. Bessler, Jane C. Hopkins

Landings, stocks and imports are given differing interactions depending on the subjective opinions of the authors. The univariate model was a thirteenth-order autoregression for each individual series. Root mean-squared errors based on i-, 2-, 6-, 12- and 18-month-ahead out-of-sample forecasts are given in Table 2. These forecasts are all over monthly observations from January, 1980 to December, 1982. Models are sequentially updated using the Kalman filter (see Doan & Litterman, 1981). From Table 2, one can see that no one method does best at all horizons for all series. The general Bayesian VAR does quite well with regard to landings at all horizons, but does rather poorly in forecasting stocks-especially at long horizons. The univariate model does a good job at all horizons at forecasting stocks, but deteriorates rather quickly in forecasts of prices. At long horizons the univariate model gives a root meansquared error two-and-one-half times as large as that from the symmetric Bayesian prior. If one must generalize from Table 2, the forecasting competition favors forecasts based on the symmetric prior vector autoregression. In only one series and at one horizon does the symmetric prior VAR perform poorly--stocks at 18-month horizons. For all other series at all other horizons, the symmetric prior is quite competitive--usually ranking first.

FINAL REMARKS In this paper we have explored the random walk prior as a restriction on the multivariate autoregression. We suggested that a random walk prior does have certain a priori appeal (efficient market hypothesis) and empirical support. A forecasting competition with data from the US shrimp market is carried out using two specifications of the random walk prior, an unrestricted vector autoregression and a univariate autoregression. The results on.five forecast horizons offer empirical support for the symmetric random walk prior. The information used for the non-symmetric prior reflects the authors' subjective information and may not reflect a particularly high level of substantive knowledge. Further research using 'real world" commodity experts may show the non-symmetric/general) prior in a better light.

Forecasting an agricultural system with random walk priors

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REFERENCES Bessler, D. A. (1980). Aggregated personalistic beliefs on yields of selected crops using ARIMA processes, American Journal of Agricuhural Economics, 62, 666--74. Bessler, D. A. (1982). Adaptive expectations, the exponentially weighted forecast, and optimal statistical predictors: A revisit, Agricultural Economics Research. 34. 16-23. Bessler, D. A. (1984a). Analysis of dynamic economic relationships: An application to the US hog market, Canadian Journal of Agricultural Economics, 32, 109-24. Bessler, D. A. (1984b). Relative prices and money: A vector autoregression on Brazilian data, American Journal of Agricultural Economics, 66, 25-30. Bessler, D. A. & Kling, J. L. (|985). Forecasting vector autoregressions with Bayesian priors, Paper presented at the Fifth International Symposium on Forecasting, Montreal, Canada, 9 June 1985. Doan, T. & Litterman, R. (1981). RATS." A user's guide, Minneapolis, VAR Econometrics. Doan. T., Litterman, R. & Sims. C. (1984). Forecasting and conditional projections, Econometric Reciews, 1, 79-89. Granger. C. W. J. (1966). The typical spectral shape of an economic variable, Econometrica, 34, 150-61, Granger, C. W. J. & Newbold, P. (1977). Forecasting economic time series. New York, Academic Press. Hopkins. Jane C. (1983). An anah'sis of the impact of alternative import management policies/or shrimp. Unpublished MS Thesis, Texas A&M University. Kling, J. & Bessler, D. A. (1985). A comparison of multivariate forecasting proecedures for economic time series, International Journal of Forecasting, 1, 5-24, Learner, E. (1976). Specification searches, New York, John Wiley. Litterman, R. (1979). Techniques for forecasting using cector autoregressions. Unpublished PhD Thesis, University of Minnesota. Plosser. C. & Schwert, W. (1978). Money, income, and sunspots: Measuring economic relationships and effects of differencing, Journal of Monetary Economics, 4, 637-60. Samuelson, P. (1965). Proof that properly anticipated prices fluctuate randomly, Industrial Management Review, 6, 41-9. Zellner, A. (1971). An introduction to Bayesian inference in econometrics, New York, John Wiley. Zellner, A. (L982). Comment: The measurement of linear dependence and feedback between multiple time series, Journal of the American Statistical Association, 77. 313-t4.