Agribusiness Management · Functional core · also AEC-505, AEC-603 Advanced Econometrics, ABM 601 Econometrics for Agri-Business, Applied Econometrics for Rural Management (IRMA)
Estimating economic relationships from data and knowing when the estimates can be trusted: OLS and its assumptions, their violations, simultaneous equations, time series and binary-choice models.
In the exams: agricultural economics PhD entrance and fellowship examinations, quantitative methods (heavy); agribusiness management PhD entrance and fellowship examinations (present)
Econometrics is the use of economic theory, mathematics and statistical inference to measure economic relationships from observed data. Theory says that demand for wheat falls when its price rises; econometrics estimates by how much, tells you how uncertain that estimate is, and warns you when the data cannot answer the question. Agricultural economists were among its first users: the identification problem was first stated clearly for farm-commodity demand and supply curves, and the partial-adjustment model of supply response was built for American field crops.
In agribusiness and agricultural economics the same handful of questions keep coming up. How does yield respond to fertiliser and irrigation? What is the price elasticity of demand for milk, onions or edible oil? How fast do farmers adjust acreage to last season's price? Are two wholesale markets integrated, so that a price shock in one is passed to the other? Which household characteristics make a farmer adopt drought-tolerant seed, crop insurance or a mobile advisory service? Each can go wrong in predictable ways: collinear inputs, error variance that grows with farm size, autocorrelated series, prices and quantities determined together, trending series that only look related, and yes/no outcomes.
This manual follows the order of a standard first graduate course (the ICAR course AEC-505, ABM 601, and their equivalents at universities in Africa, Europe, the Americas and Asia): the classical linear regression model and ordinary least squares, the Gauss-Markov theorem and maximum likelihood, hypothesis testing, dummy variables, the three classic violations (multicollinearity, heteroscedasticity, autocorrelation), specification error, distributed lags, simultaneous equations, time series (stationarity, unit roots, ARIMA, cointegration and error correction) and binary and censored dependent variables. Notation follows the common textbooks: Y is the dependent variable, X the regressors, u the population error, e the OLS residual, n the number of observations and k the number of estimated coefficients including the intercept.
Examinations test it at three depths: Master's entrance papers ask definitions and interpretations; PhD entrance and lecturer-eligibility papers ask for the consequences of each violation, the logic of each test, identification and short computations; practitioners are judged on choosing a model the data can support and saying honestly what an estimate shows.
A regression studies how the conditional mean of a dependent variable Y changes with one or more explanatory variables X. The population regression function (PRF) for two variables is E(Y|X) = b1 + b2X, and an individual observation is Yi = b1 + b2Xi + ui, where ui is the stochastic disturbance or error term. It stands for everything that affects Y but is not in the model: omitted variables such as soil quality, weather, and measurement error in Y. Because we never observe the population, we estimate a sample regression function (SRF), Yi = b1-hat + b2-hat Xi + ei, where ei is the residual. "Linear" in the classical model means linear in the parameters, not necessarily in the variables: Y = b1 + b2 ln X, ln Y = b1 + b2 ln X and Y = b1 + b2(1/X) are all linear regression models, while Y = b1 + b2^2 X is not linear in parameters.
Ordinary least squares (OLS) chooses the estimates that minimise the residual sum of squares, RSS = sum of ei squared. Setting the partial derivatives to zero gives the normal equations, whose solution in the two-variable case is b2-hat = sum(xi yi) / sum(xi squared), where lower-case letters are deviations from the sample means (xi = Xi - X-bar), and b1-hat = Y-bar - b2-hat X-bar. Several algebraic properties follow and hold in every sample with an intercept: the fitted line passes through the point of means; the residuals sum to zero; the residuals are uncorrelated with each regressor and with the fitted values; and the mean of the fitted values equals the mean of Y. For example, if a sample of 10 wheat plots gives sum(xy) = 480 and sum(x squared) = 1,200 in deviation form for nitrogen (kg/ha) and yield (q/ha), the slope is 0.4 quintal per kg of nitrogen. With mean nitrogen 50 and mean yield 30, the intercept is 30 - 0.4 x 50 = 10 quintals: the predicted yield with no nitrogen, taken seriously only if zero nitrogen lies within or near the observed range.
The classical linear regression model (CLRM), as set out by Gujarati and others, rests on a list of assumptions. (1) The model is linear in parameters and correctly specified. (2) The X values are fixed in repeated sampling, or, in the modern version, independent of the error. (3) The error has zero conditional mean: E(ui|Xi) = 0. (4) Homoscedasticity: var(ui|Xi) = sigma squared, the same for every observation. (5) No autocorrelation: cov(ui, uj|X) = 0 for i not equal to j. (6) The number of observations exceeds the number of parameters. (7) There is variation in each X, and no exact linear relationship among the regressors (no perfect multicollinearity). For inference in small samples a further assumption is added: (8) ui is normally distributed. Assumption 3 is the one that matters most for unbiasedness, and it fails whenever a relevant variable correlated with X is left out, when X is measured with error, or when X and Y are determined simultaneously.
The total variation in Y, TSS = sum(yi squared), splits into the explained sum of squares ESS = sum(y-hat squared) and RSS. The coefficient of determination R-squared = ESS/TSS = 1 - RSS/TSS lies between 0 and 1 when an intercept is included, and in the two-variable case equals the squared correlation between X and Y. R-squared never falls when a regressor is added, so the adjusted R-squared, 1 - (1 - R-squared)(n - 1)/(n - k), is used to compare models with different numbers of regressors on the same dependent variable; it can be negative. R-squared cannot be compared across models whose dependent variables differ (Y against ln Y), and a high R-squared is not evidence of a correct model: two trending series often give R-squared above 0.9 with no real relationship. Functional form decides what the slope means. In the linear model the slope is a constant marginal effect and the elasticity varies along the line; it is usually reported at the means, b2 times X-bar/Y-bar. In the log-log (double-log) model ln Y = b1 + b2 ln X the slope is itself the constant elasticity, which is why Cobb-Douglas production and constant-elasticity demand functions are estimated this way. In the log-lin model ln Y = b1 + b2 t, b2 is the instantaneous growth rate and the compound annual growth rate is (antilog of b2 - 1) x 100 percent, the standard way agricultural economists compute growth in area, production and yield. In the lin-log model Y = b1 + b2 ln X, b2/100 is the absolute change in Y for a one percent change in X; the elasticity is b2/Y.
An estimator is unbiased if its sampling distribution is centred on the true value, efficient if it has the smallest variance in a stated class, and consistent if it converges to the true value as n grows; mean squared error equals variance plus squared bias. Consistency is the minimum requirement. The Gauss-Markov theorem states that, under the CLRM assumptions (linearity in parameters, exogenous regressors with zero conditional mean of the error, homoscedasticity, no autocorrelation and no perfect multicollinearity), the OLS estimators have the minimum variance in the class of all linear unbiased estimators: they are BLUE, the best linear unbiased estimators. Three points are regularly tested. First, normality of the error is not needed for the theorem; it is needed only for exact t and F distributions in small samples. Second, "best" is relative to linear unbiased estimators only; a biased or non-linear estimator can have a smaller variance. Third, each assumption plays a distinct role: zero conditional mean delivers unbiasedness, while homoscedasticity and no autocorrelation deliver the minimum variance. That is why heteroscedasticity and autocorrelation leave OLS unbiased but not efficient, whereas omitted variables and simultaneity make it biased. In the two-variable model, var(b2-hat) = sigma squared / sum(x squared), so the slope is estimated more precisely when the error variance is small, the sample is large and X is widely spread. Sigma squared is estimated without bias by RSS/(n - k).
Maximum likelihood (ML) chooses the parameter values under which the observed sample would have been most probable. You write the joint density of the sample as a function of the parameters (the likelihood), take its logarithm and maximise. For the linear model with normal, homoscedastic, non-autocorrelated errors, the log-likelihood is -(n/2) ln(2 pi) - (n/2) ln sigma squared - RSS/(2 sigma squared). Maximising over the coefficients means minimising RSS, so the ML estimators of the coefficients are identical to OLS. The ML estimator of the error variance is RSS/n, which is biased downward in small samples (OLS divides by n - k) but consistent, since the two differ by a factor that goes to one. Under standard regularity conditions ML estimators are consistent, asymptotically normal and asymptotically efficient, reaching the Cramer-Rao lower bound given by the inverse of the information matrix, and they are invariant: the ML estimate of a function of a parameter is that function of the ML estimate. ML is how logit, probit, Tobit, ARIMA and full-information simultaneous-equation models are estimated. Three asymptotically equivalent tests come from the likelihood: the likelihood ratio test, LR = 2(lnL unrestricted - lnL restricted), chi-square with degrees of freedom equal to the number of restrictions; the Wald test, which needs only the unrestricted model; and the Lagrange multiplier (score) test, which needs only the restricted model, which is why many diagnostic tests are LM tests of the n R-squared form. In the linear model W >= LR >= LM.
With normal errors, (b-hat - b)/se(b-hat) follows Student's t distribution with n - k degrees of freedom. To test H0: b2 = 0 you compute t = b2-hat/se(b2-hat) and compare it with the critical value; a common rule of thumb, the "2-t" rule, says that with more than about 20 degrees of freedom a t above 2 in absolute value is significant at roughly the 5 percent level in a two-tailed test. To test a value other than zero, such as whether a demand elasticity equals -1, use t = (b-hat - (-1))/se. A 100(1 - alpha) percent confidence interval is b-hat plus or minus t(alpha/2, n - k) x se; a two-tailed test rejects the hypothesised value exactly when it lies outside the interval. The p-value is the lowest significance level at which H0 would be rejected. Two distinctions are examined often. Statistical significance is not economic significance: a precisely estimated milk-price effect of 0.001 may be significant and irrelevant. And failing to reject is not accepting: a wide interval that includes zero also includes large effects.
The overall F test asks whether all slope coefficients are jointly zero. From the analysis-of-variance table, F = [ESS/(k - 1)] / [RSS/(n - k)], with (k - 1, n - k) degrees of freedom, and in terms of R-squared, F = [R-squared/(k - 1)] / [(1 - R-squared)/(n - k)]. The two forms are identical, so a test of F is also a test that the population R-squared is zero. In the two-variable model F equals t squared for the slope. Individual t tests and the joint F test can disagree: with collinear regressors each t can be insignificant while F is highly significant, because the data show that the regressors together matter but cannot separate them. For example, a milk-yield equation with n = 30 and k = 4 and R-squared = 0.60 gives F = (0.60/3)/(0.40/26) = 13.0, far above the 5 percent critical value of about 2.98 for (3, 26) degrees of freedom.
Any set of linear restrictions (for example that two coefficients are equal, or that the elasticities in a Cobb-Douglas function sum to one, meaning constant returns to scale) can be tested by estimating the model twice. The unrestricted model gives RSS-U; the model with the restrictions imposed gives RSS-R, which can never be smaller. Then F = [(RSS-R - RSS-U)/m] / [RSS-U/(n - k)], where m is the number of restrictions and k the number of coefficients in the unrestricted model. When the dependent variable is the same in both models, the equivalent R-squared form is F = [(R-squared-U - R-squared-R)/m] / [(1 - R-squared-U)/(n - k)]. A single restriction can also be tested by a t test on the combined coefficient, using the variances and covariance of the estimates: to test b2 + b3 = 1, t = (b2 + b3 - 1)/sqrt[var(b2) + var(b3) + 2cov(b2, b3)]. Leaving out the covariance term is a common mistake.
A relationship estimated over a long period, or over two regions, may not be the same throughout: acreage response may have changed after a market reform, or input response may differ between irrigated and rainfed farms. The Chow test compares a pooled regression over all n1 + n2 observations (RSS-R) with separate regressions on each sub-sample, whose residual sums add to RSS-U. Then F = [(RSS-R - RSS-U)/k] / [RSS-U/(n1 + n2 - 2k)], with (k, n1 + n2 - 2k) degrees of freedom. The test assumes that the error variance is the same in both sub-samples (checked first, for example with a Goldfeld-Quandt style F ratio), that the break date is known in advance, and it tells you that the relationship changed but not whether it was the intercept, the slope or both. When a sub-sample has fewer observations than coefficients, the Chow predictive test is used. The dummy-variable approach in the next chapter shows which parameter changed.
Many things that matter in agriculture are qualitative: irrigated or rainfed, member of a cooperative or not, season, region, variety, gender of the household head, before or after a policy change. A dummy (indicator) variable takes the value 1 if the attribute is present and 0 otherwise. A variable with m categories needs m - 1 dummies when the regression has an intercept. The category with no dummy is the base or reference category, and each dummy coefficient measures the difference in the intercept between that category and the base. Including all m dummies with an intercept creates perfect multicollinearity, because the dummies sum to the intercept column; this is the dummy variable trap, and OLS cannot be computed. The alternatives are m - 1 dummies with an intercept, or m dummies with no intercept, in which case each coefficient is the mean for that category directly. A regression of Y on dummies alone is an analysis-of-variance (ANOVA) model; one with both dummies and continuous regressors is an analysis-of-covariance (ANCOVA) model. For quarterly price data with an intercept, three seasonal dummies are used; for four agro-climatic zones, three zone dummies.
In Yi = b1 + b2Di + b3Xi + ui, the dummy shifts the intercept only: irrigated farms have the same response to fertiliser but a different yield level. Adding an interaction term DiXi gives Yi = b1 + b2Di + b3Xi + b4(DiXi) + ui. For the base group the line is b1 + b3X; for the dummy group it is (b1 + b2) + (b3 + b4)X. Here b2 is the differential intercept and b4 the differential slope. Testing b2 = b4 = 0 jointly is equivalent to the Chow test, but the dummy approach also shows which parameter changed and uses a single pooled regression with more degrees of freedom. The interaction of two dummies (for example irrigated and improved variety) tests whether the effect of one attribute depends on the other. Piecewise (spline) regression uses a dummy to let the slope change at a known threshold, such as a farm-size cut-off for a subsidy.
When the dependent variable is in logarithms, a dummy coefficient c is not an exact percentage effect. The proportional difference between the two groups is (e^c - 1); for small c it is close to c, but for large c the gap matters. A coefficient of 0.30 on an irrigation dummy in a log-yield equation implies irrigated yields about 35 percent higher (e^0.30 - 1 = 0.350), not 30 percent. Dummies are also used for seasonal adjustment of monthly or quarterly price series (regress on seasonal dummies and keep the residuals plus the mean), for outliers or one-off events such as a drought year or an export ban, for policy-period shifts, and for fixed effects in panel data, where one dummy for each farm or district (less one) absorbs all time-invariant differences between units. When the dependent variable itself is a dummy the model is a binary-choice model, treated in the last chapter.
Multicollinearity is a linear relationship among the regressors. Perfect multicollinearity (one regressor is an exact linear combination of others) makes X'X singular, so the coefficients cannot be estimated; it arises from mistakes such as the dummy trap or including total area together with irrigated and unirrigated area. The practical problem is near multicollinearity. In farm data it is everywhere: fertiliser, seed, labour and machinery use all rise with farm size; rainfall and irrigation move together; prices of substitute commodities trend together over time. Near multicollinearity does not violate any CLRM assumption, so OLS remains BLUE. The trouble is that "best" may still be poor. In a two-regressor model var(b2) = sigma squared / [sum(x2 squared)(1 - r23 squared)], where r23 is the correlation between the two regressors, so the variance rises without limit as r23 approaches one. The consequences: large standard errors and wide confidence intervals; individual t ratios that are insignificant even though R-squared is high and the F test is significant; coefficients that are sensitive to small changes in the data or specification and may take theoretically wrong signs.
There is no single test; the usual indicators are these. A high R-squared with few significant t ratios. High pairwise correlations among regressors, above about 0.8, though collinearity can involve three or more variables with modest pairwise correlations. Auxiliary regressions of each regressor on the others: Klein's rule of thumb says multicollinearity is troublesome if the R-squared of an auxiliary regression exceeds the overall R-squared. The variance inflation factor, VIF(j) = 1/(1 - R-squared j), where R-squared j comes from regressing Xj on the other regressors; VIF above 10 (R-squared j above 0.90) is the common warning level. Tolerance is 1/VIF. Eigenvalues of X'X: the condition number k = maximum eigenvalue/minimum eigenvalue and the condition index = its square root; a condition index between 10 and 30 indicates moderate and above 30 severe collinearity in the Belsley-Kuh-Welsch guideline. The Farrar-Glauber procedure adds chi-square, F and t tests for the presence, location and pattern of collinearity.
The first remedy is often to do nothing: if the aim is prediction and the collinear pattern will continue, or if the key coefficients are significant anyway, collinearity is harmless. Otherwise the options are these. Use a priori information, for example imposing constant returns to scale in a Cobb-Douglas function, which reduces the number of free parameters. Combine cross-section and time-series information, as Tobin did by taking the income elasticity from household budget data and then estimating the price elasticity from time series. Drop a variable, but this risks the omitted-variable bias of the specification chapter, which is usually worse than imprecision. Transform variables: first differences or ratios (per-hectare values) can reduce collinearity but may introduce autocorrelation or heteroscedasticity. Collect more data or data with more spread. Use principal components regression or ridge regression, which trade a small bias for a large reduction in variance; ridge estimates are biased, and principal components can be hard to interpret. What you should not do is drop a variable only because its t ratio is low when theory says it belongs.
Heteroscedasticity means that the variance of the error differs across observations: var(ui) = sigma-i squared. It is typical of cross-section data where units differ in size. Output, costs and income vary more among large farms than among small ones; food expenditure varies more among rich households than poor ones; error-learning models suggest variance falls as farmers gain experience; outliers, skewed regressors and a wrong functional form also produce it. Under heteroscedasticity OLS coefficient estimates remain unbiased and consistent, because the zero-mean assumption still holds, but they are no longer efficient, so they are not BLUE. More serious in practice, the usual OLS formula for the variances is wrong, so standard errors, t ratios, F tests and confidence intervals are unreliable. The direction of the bias in the standard errors depends on how the variance is related to the regressors; when the variance rises with X, the conventional standard errors typically understate the true sampling variability of the slope.
Informal methods plot squared residuals against the fitted values or against each regressor and look for a fan or funnel. Formal tests: the Park test regresses ln(e squared) on ln X and tests the slope; the Glejser test regresses the absolute residual on X, its square root or reciprocal. The Spearman rank-correlation test correlates the ranks of the absolute residuals with the ranks of X. The Goldfeld-Quandt test orders the observations by the variable thought to drive the variance, omits c central observations, runs separate regressions on the two ends and computes F = RSS2/RSS1 (larger-variance group on top) with (n - c - 2k)/2 degrees of freedom in both numerator and denominator. The Breusch-Pagan-Godfrey test regresses the squared residuals (scaled by their mean) on a set of variables Z and uses half the explained sum of squares, which is chi-square with degrees of freedom equal to the number of Z variables. White's general test regresses the squared residuals on the regressors, their squares and their cross-products, and uses n x R-squared, chi-square with degrees of freedom equal to the number of regressors in the auxiliary regression excluding the constant.
If the form of the variance is known, generalised least squares (GLS) is BLUE. With var(ui) = sigma squared times Xi squared, dividing every term of the equation by Xi gives a transformed model with a homoscedastic error; OLS on the transformed data is weighted least squares (WLS), each observation weighted inversely to its error variance. With var(ui) proportional to Xi, divide by the square root of Xi. When the form is unknown, feasible GLS estimates it from an auxiliary regression. The most widely used practical remedy today is to keep the OLS coefficients and use White's heteroscedasticity-consistent (robust) standard errors, which are valid in large samples whatever the form. Re-specifying the model can also help: a log transformation compresses the scale, and expressing variables per hectare or per household member removes much size-driven heteroscedasticity. If WLS and OLS estimates differ a great deal, suspect specification error rather than heteroscedasticity alone.
Autocorrelation (serial correlation) is correlation between error terms of different observations, most commonly successive periods in time-series data. Sources include inertia in economic series, omitted variables that are themselves autocorrelated, wrong functional form, the cobweb phenomenon in agricultural supply (this year's acreage responds to last year's price, so errors in planting decisions carry over), data smoothing and interpolation (quarterly figures built from annual ones), and nonstationarity. The standard model is the first-order autoregressive scheme AR(1): ut = rho u(t-1) + et, where -1 < rho < 1 and et is white noise. Under autocorrelation OLS remains unbiased and consistent, provided no lagged dependent variable is among the regressors, but it is not efficient. The conventional variance formula is wrong: with positive autocorrelation and a positively autocorrelated regressor, it usually underestimates the true variances, so t ratios and R-squared are inflated and variables look more significant than they are. If the regressors include Y(t-1) and the errors are autocorrelated, OLS becomes biased and inconsistent.
The Durbin-Watson statistic is d = sum from t = 2 to n of (et - e(t-1)) squared divided by sum from t = 1 to n of et squared. For large samples d is approximately 2(1 - rho-hat), so d is near 2 with no first-order autocorrelation, near 0 with strong positive autocorrelation and near 4 with strong negative autocorrelation. Because the exact distribution depends on X, Durbin and Watson tabulated a lower bound dL and an upper bound dU. For positive autocorrelation: reject if d < dL, do not reject if d > dU, and the test is inconclusive between them. For negative autocorrelation the mirror rules use 4 - dU and 4 - dL. The test assumes an intercept in the model, non-stochastic regressors, AR(1) errors, no missing observations and, crucially, no lagged dependent variable among the regressors; with Y(t-1) present, d is biased toward 2 and Durbin's h statistic is used instead: h = rho-hat times the square root of n/[1 - n var(coefficient of Y(t-1))], approximately standard normal, and not computable if n times that variance is 1 or more. A useful warning sign: in a levels regression of trending series, R-squared greater than d suggests a spurious regression.
The Breusch-Godfrey (BG) test is a Lagrange multiplier test that allows higher-order AR or MA errors and lagged dependent variables: regress the OLS residuals on the original regressors and p lagged residuals, and use (n - p) R-squared, chi-square with p degrees of freedom. Before correcting, ask whether the autocorrelation is pure or is caused by an omitted variable or wrong functional form, in which case the cure is to fix the model. For pure AR(1) autocorrelation with rho known, the generalised difference equation (Yt - rho Y(t-1)) = b1(1 - rho) + b2(Xt - rho X(t-1)) + et has a white-noise error and OLS on it is GLS; the Prais-Winsten transformation keeps the first observation by multiplying it by the square root of (1 - rho squared). With rho equal to 1 the transformation becomes the first-difference model with no intercept. When rho is unknown it is estimated from d (rho-hat = 1 - d/2), from the residuals, or iteratively by the Cochrane-Orcutt procedure; these are feasible GLS. As with heteroscedasticity, one can instead keep OLS and use Newey-West heteroscedasticity-and-autocorrelation-consistent (HAC) standard errors, which are valid in large samples.
Specification error means the estimated model differs from the true one. The most damaging kind is omitting a relevant variable. If the true model is Y = b1 + b2X2 + b3X3 + u and X3 is left out, the slope from regressing Y on X2 alone has expected value b2 + b3 b32, where b32 is the slope from regressing the omitted X3 on the included X2. The estimator is therefore biased and inconsistent unless b3 = 0 or X2 and X3 are uncorrelated; the error variance is estimated with bias, and the usual confidence intervals and tests are misleading. A classic agricultural example: estimating the return to fertiliser without controlling for soil quality or irrigation, both of which raise fertiliser use and yield, overstates the fertiliser effect. Including an irrelevant variable is much less harmful: the estimates remain unbiased and consistent, and the error variance is correctly estimated, but the variances are larger, so the estimates are inefficient. The asymmetry explains the usual advice to err on the side of including variables theory suggests, while guarding against overfitting.
Choosing a linear form when the true relationship is logarithmic or quadratic is also a specification error and shows up as patterned residuals, apparent heteroscedasticity or autocorrelation. Measurement error behaves differently depending on where it sits. Error in the dependent variable is absorbed into the disturbance: OLS stays unbiased but less precise. Error in an explanatory variable is serious: the mismeasured X is correlated with the composite error, so OLS is biased and inconsistent, and in the two-variable case with classical errors the slope is biased toward zero (attenuation bias) by the reliability ratio var(X*)/[var(X*) + var(measurement error)]. The standard remedy is instrumental variables (IV): find a variable correlated with the true X but uncorrelated with the error, such as a second independent measurement or a GPS-measured area as an instrument for a farmer-reported one. Recalled input quantities, self-reported plot areas and household income are all prone to such error.
Look first at the residuals: patterns against fitted values or omitted candidates suggest misspecification. Ramsey's RESET test adds powers of the fitted values (Y-hat squared, Y-hat cubed) to the regression and tests their joint significance with an F test; a significant result says the model is misspecified but not how. The Lagrange multiplier test for added variables regresses the restricted model's residuals on all candidate regressors and uses n R-squared, chi-square with degrees of freedom equal to the number of added variables. To choose between a linear and a log-linear model, the MacKinnon-White-Davidson (MWD) test adds the difference between the fitted values of one form and the exponentiated fitted values of the other and tests its significance. For non-nested models, where neither model is a special case of the other, the Davidson-MacKinnon J test adds the fitted values of the rival model as a regressor; it can reject both models or neither. The Hausman test compares an estimator that is consistent under both hypotheses (such as IV) with one that is efficient under the null (OLS); a large difference signals endogeneity.
When models have the same dependent variable, several criteria trade fit against parsimony. Adjusted R-squared penalises extra regressors only slightly. Akaike's information criterion, AIC, is commonly written ln(RSS/n) + 2k/n; Schwarz's criterion, SIC or BIC, is ln(RSS/n) + k ln(n)/n. Both are chosen to be minimised, and because ln(n) exceeds 2 once n is 8 or more, SIC penalises extra parameters more heavily and tends to pick smaller models. These criteria are also used to choose lag lengths in distributed lag, ADF and ARIMA models. Two broad philosophies exist. The traditional approach starts from a simple model and adds variables (specific-to-general); Hendry's London School of Economics approach starts from a general model with generous lags and tests down to a parsimonious one that passes diagnostic tests (general-to-specific), on the grounds that testing up from a misspecified model makes all tests invalid. Trying many specifications and reporting only the best (data mining) makes nominal significance levels meaningless.
Economic responses take time. A higher price this season affects next season's planted area; a new irrigation scheme raises output over several years; farmers' expectations of price are formed from past prices; tree crops and livestock herds respond over many years. A distributed lag model includes current and past values of X: Yt = a + b0Xt + b1X(t-1) + b2X(t-2) + ... + ut. The coefficient b0 is the short-run or impact multiplier, the partial sums b0 + b1 + ... + bi are interim multipliers, and the sum of all the b's is the long-run or total multiplier. A model that includes Y(t-1) as a regressor is autoregressive (dynamic). Estimating a long free distributed lag by OLS runs into problems: there is no rule for how long the lag is, each lag costs a degree of freedom, successive lags of X are highly collinear, and specification searches over lag length are a form of data mining. The ad hoc approach (Alt and Tinbergen) adds lags one at a time until a coefficient changes sign or becomes insignificant. A related idea is Granger causality: X "Granger-causes" Y if lagged X improves the prediction of Y beyond lagged Y alone, tested by an F test on the lagged X terms in both directions, on stationary data. It is predictive precedence, not causation; it is used, for example, to ask whether a central wholesale market leads regional markets.
Koyck assumed the lag coefficients decline geometrically: bk = b0 lambda^k, with 0 < lambda < 1, where lambda is the rate of decline and 1 - lambda the speed of adjustment. The infinite lag then collapses to Yt = a(1 - lambda) + b0Xt + lambda Y(t-1) + vt, where vt = ut - lambda u(t-1). Only three parameters remain. The long-run multiplier is b0/(1 - lambda), the mean lag is lambda/(1 - lambda) and the median lag, the time taken for half the total effect, is -ln 2/ln lambda. The transformation has a price. Y(t-1) is stochastic and correlated with vt, because both contain u(t-1), so OLS is biased and inconsistent; vt is a moving average and therefore autocorrelated; and the Durbin-Watson test is not valid, so Durbin's h is used. Instrumental variables, with X(t-1) as an instrument for Y(t-1) (Liviatan's approach), give consistent estimates. The Almon polynomial lag is used instead when lag weights are expected to rise and then fall.
Two behavioural models lead to the same estimating equation as Koyck. In the adaptive expectations model (Cagan, Friedman), Y depends on the expected price X*, and expectations are revised by a fraction gamma of the last error: X*t - X*(t-1) = gamma(Xt - X*(t-1)), with 0 < gamma <= 1. In the partial adjustment or stock adjustment model, associated with Nerlove's work on the supply of American field crops, the desired level Y*t = b0 + b1Xt is reached only gradually: Yt - Y(t-1) = delta(Y*t - Y(t-1)), where delta is the coefficient of adjustment. Substituting gives Yt = delta b0 + delta b1 Xt + (1 - delta) Y(t-1) + delta ut. The error delta ut is not autocorrelated if ut is not, so OLS is consistent in the pure partial adjustment model, unlike the Koyck and adaptive expectations forms, whose errors are moving averages. The Nerlovian supply response model combines both, with acreage as the dependent variable and lagged price, rainfall and other shifters as regressors. If At = a + b P(t-1) + c A(t-1) in logarithms, b is the short-run price elasticity, 1 - c is the adjustment coefficient, and the long-run elasticity is b/(1 - c).
In many markets variables are determined jointly. The price and quantity of an agricultural commodity come from the intersection of demand and supply; farm income and consumption influence each other; input use and yield may be chosen together. A simultaneous-equation model has several structural equations, each describing one behavioural or technical relationship. Variables determined inside the system are endogenous; those determined outside are exogenous. Exogenous variables and lagged endogenous variables together are called predetermined. Because an endogenous variable appearing as a regressor in one equation depends on the error of that equation through the rest of the system, it is correlated with the error. OLS applied to such an equation is biased and inconsistent: this is simultaneity bias (or simultaneous-equation bias). The Hausman specification test checks whether a variable can be treated as exogenous. The exception is a recursive (triangular) model, in which the first equation contains only predetermined variables, the second adds the first endogenous variable and so on, with errors uncorrelated across equations; there OLS is appropriate equation by equation.
The reduced form expresses each endogenous variable in terms of the predetermined variables and errors only. Its coefficients are functions of the structural coefficients and can be estimated consistently by OLS, because the predetermined variables are uncorrelated with the reduced-form errors. The identification problem asks whether the structural coefficients can be recovered from the reduced-form ones. Working's 1927 article on statistical demand curves made the point with farm data: if only price and quantity are observed and both curves shift randomly, the scatter of equilibrium points traces neither curve. An equation is identified if its parameters can be uniquely obtained from the reduced form. It is under-identified (not identified) if they cannot, exactly identified if there is a unique solution, and over-identified if there is more than one set of values. Identification comes from exclusion restrictions: variables that shift one curve but not the other. Rainfall or input prices shift the supply of a crop but not demand, so they identify the demand curve; consumer income shifts demand but not supply, so it identifies the supply curve.
Let M be the number of endogenous variables in the model (equal to the number of equations), m the number of endogenous variables in a given equation, K the number of predetermined variables in the model and k the number in the given equation. The order condition, a necessary condition, is that the number of predetermined variables excluded from the equation be at least the number of endogenous variables in it minus one: K - k >= m - 1. If K - k = m - 1 the equation is exactly identified; if K - k > m - 1 it is over-identified. An equivalent statement is that the number of variables (endogenous plus predetermined) excluded from the equation must be at least M - 1. The order condition is not sufficient, because the excluded variables might be excluded from every equation or appear in a proportional pattern. The rank condition is necessary and sufficient: in a model of M equations, an equation is identified if and only if at least one non-zero determinant of order (M - 1) x (M - 1) can be formed from the coefficients of the variables excluded from that equation but included in the other equations. Consider demand Qd = a0 + a1P + a2Y + u1 and supply Qs = b0 + b1P + b2R + u2, where Y is income and R rainfall. M = 2, K = 2. Demand excludes R: K - k = 1 = m - 1, so it is exactly identified; supply excludes Y, and is also exactly identified.
Single-equation (limited-information) methods estimate one equation at a time. Indirect least squares (ILS) estimates the reduced form by OLS and solves back for the structural coefficients; it applies only to exactly identified equations and gives consistent but biased estimates. Two-stage least squares (2SLS), developed by Theil and Basmann, first regresses each right-hand endogenous variable on all predetermined variables in the system and then uses the fitted values in place of the endogenous variable in the second stage. It works for exactly identified and over-identified equations; in the exactly identified case it gives the same estimates as ILS. It is a form of instrumental variables estimation, and the standard errors must be computed from the correct residuals, not taken from a naive second-stage OLS. System (full-information) methods estimate all equations together, using the correlation of errors across equations: three-stage least squares (3SLS, Zellner and Theil) and full-information maximum likelihood (FIML). They are more efficient if the whole system is correctly specified, but an error in one equation spreads to the others. Under-identified equations cannot be estimated by any method.
A time series is (weakly or covariance) stationary if its mean and variance are constant over time and the covariance between two values depends only on the lag between them, not on the date. Most agricultural prices, outputs and input-use series in levels are not stationary: they trend or wander. The simplest non-stationary process is the random walk, Yt = Y(t-1) + ut, whose variance t sigma squared grows with time; a random walk with drift adds a constant, Yt = d + Y(t-1) + ut. Its first difference is stationary, so it is integrated of order one, I(1). A series that needs d differences to become stationary is I(d); a stationary series is I(0). A trend-stationary series, Yt = a + bt + ut with stationary ut, becomes stationary by removing a deterministic trend; a difference-stationary series needs differencing. Treating one as the other is a mistake: detrending a random walk leaves a non-stationary residual. The danger of non-stationarity is spurious regression, shown by Granger and Newbold: regressing one independent random walk on another typically gives a high R-squared, significant t ratios and a very low Durbin-Watson statistic, although the series are unrelated. Standard t and F distributions do not apply to regressions of I(1) variables in levels unless they are cointegrated.
The Dickey-Fuller (DF) test writes Yt = rho Y(t-1) + ut, subtracts Y(t-1) from both sides to get Delta Yt = delta Y(t-1) + ut with delta = rho - 1, and tests H0: delta = 0 (a unit root, non-stationary) against H1: delta < 0 (stationary). Under the null the t ratio on delta does not follow the t distribution, even in large samples; it is compared with the tau critical values tabulated by Dickey and Fuller and refined by MacKinnon, which are more negative than the usual t values. Three versions exist: with no constant, with a constant (drift), and with a constant and a linear trend; each has its own critical values, and the choice should follow a plot of the data. The augmented Dickey-Fuller (ADF) test adds lagged differences, Delta Y(t-i), to remove autocorrelation from the error; lag length is chosen by an information criterion. Reject the unit root only when the computed tau is more negative than the critical value. The Phillips-Perron test corrects for autocorrelation non-parametrically instead. The KPSS test reverses the hypotheses, taking stationarity as the null, and is often run alongside ADF as a check. Unit root tests have low power against near-unit-root alternatives in short samples, and a structural break, such as a policy change or a drought, can make a stationary series look non-stationary (Perron).
An autoregressive process of order p, AR(p), expresses Yt as a function of its own p past values plus white noise; a moving-average process of order q, MA(q), expresses it as a weighted sum of the current and q past white-noise shocks. ARMA(p, q) combines them, and ARIMA(p, d, q) applies ARMA to a series differenced d times. AR(1), Yt = c + phi Y(t-1) + et, is stationary if the absolute value of phi is less than one; its mean is then c/(1 - phi). An MA process is always stationary but must satisfy invertibility conditions. Box and Jenkins proposed four steps. Identification: difference until stationary to fix d, then use the autocorrelation function (ACF) and partial autocorrelation function (PACF). For a pure AR(p) the ACF decays gradually and the PACF cuts off after lag p; for a pure MA(q) the ACF cuts off after lag q and the PACF decays; for a mixed ARMA both decay. Estimation, usually by maximum likelihood. Diagnostic checking: the residuals should be white noise, tested with the Box-Pierce Q = n sum of squared residual autocorrelations or the Ljung-Box LB = n(n + 2) sum of rho-k squared/(n - k), both chi-square. Forecasting. Seasonal ARIMA adds seasonal AR and MA terms and seasonal differencing for monthly or quarterly data such as vegetable prices. ARIMA models are atheoretical: they forecast well in the short run but do not explain.
Two I(1) series are cointegrated if some linear combination of them is I(0): they may wander, but they wander together because an economic force ties them, such as spatial arbitrage between two grain markets or the link between farm-gate and retail prices. A regression of cointegrated series in levels is not spurious, and the OLS estimate of the cointegrating coefficient is super-consistent, converging faster than usual. The Engle-Granger two-step method first estimates the levels regression Yt = b1 + b2Xt + ut by OLS and then applies an ADF test to the residuals, using Engle-Granger critical values, which are more negative than ordinary DF values because the residuals come from an estimated regression. The Johansen procedure, based on maximum likelihood in a vector autoregression, tests the number of cointegrating vectors with the trace and maximum-eigenvalue statistics; with n variables there can be at most n - 1 cointegrating vectors. The Granger representation theorem says that if two variables are cointegrated, their short-run relationship can be written as an error-correction model (ECM): Delta Yt = a0 + a1 Delta Xt + a2 u(t-1) + et, where u(t-1) is last period's deviation from the long-run equilibrium. The coefficient a2 should be negative and between 0 and -1 in absolute terms: it is the fraction of the disequilibrium corrected each period, so a2 = -0.4 means 40 percent of a gap between two markets' prices is closed within one period, and the half-life of a shock is ln(0.5)/ln(1 + a2) periods. Market-integration and price-transmission studies in agricultural marketing rest on exactly this machinery.
Many agricultural decisions are yes/no: adopt a hybrid or not, insure or not, sell through a farmer producer organisation or a trader, take formal credit or not. With a dependent variable that is 1 or 0, E(Y|X) is the probability that Y = 1. The linear probability model (LPM) simply regresses Y on X by OLS, so a slope is the change in the probability for a unit change in X. It is easy to estimate and interpret but has four well-known problems: the error is not normal (it takes only two values for given X); it is heteroscedastic, with variance p(1 - p) that changes with X; fitted probabilities can fall below 0 or above 1; and the marginal effect is constant, while in reality a change in X usually matters most in the middle of the probability range and little near 0 or 1. The conventional R-squared is also of limited value.
Logit and probit keep predicted probabilities between 0 and 1 by passing a linear index Z = b1 + b2X2 + ... through a cumulative distribution function. In the logit model P = 1/(1 + e^-Z), the logistic CDF, so the log of the odds, ln[P/(1 - P)] = Z, is linear in the regressors. A logit slope is the change in the log-odds for a unit change in X, and e^b is the odds ratio. In the probit model P = the standard normal CDF of Z. Both are estimated by maximum likelihood, not OLS. Because the models are non-linear, the marginal effect of X on the probability is not the coefficient: for logit it is b times P(1 - P), largest at P = 0.5 where it equals b/4; for probit it is b times the standard normal density at Z. Marginal effects are reported at the means of the regressors or averaged over the sample. The logistic distribution has slightly fatter tails, but the two models usually give very similar fitted probabilities and marginal effects; logit coefficients are roughly 1.6 to 1.8 times the corresponding probit coefficients because the standard logistic variance is pi squared/3 rather than 1. Goodness of fit is judged by McFadden's pseudo R-squared, 1 - lnL(model)/lnL(intercept only), by the percentage of observations correctly predicted, and by the likelihood ratio test of all slopes, which is chi-square with degrees of freedom equal to the number of slopes.
Some outcomes are continuous but piled up at a limit. Many farm households apply no chemical fertiliser, buy no insurance or sell nothing in a given season, so the quantity is zero for a large share and positive for the rest. The Tobit model, proposed by Tobin in 1958 for household spending on durables, treats the observed Y as max(0, Y*), where Y* = Xb + u is a latent variable with a normal error. OLS on the full sample, or on the positive observations only, gives biased and inconsistent estimates; Tobit is estimated by maximum likelihood, combining a probit-like part for the zeros and a normal regression part for the positive values. A Tobit coefficient is the effect on the latent Y*, and McDonald and Moffitt showed how the total effect on the observed Y splits into a change in the probability of being above the limit and a change in the amount given that the unit is above it. Tobit assumes the same variables and coefficients drive both the decision to participate and the amount; when they may differ (for example, the decision to adopt irrigation may depend on water access while the area irrigated depends on labour), the Cragg double-hurdle model is used. Heckman's two-step procedure addresses sample selection, where the outcome is observed only for a selected group (wages only for those who work off-farm): a probit for selection yields the inverse Mills ratio, which is added as a regressor in the outcome equation.
A Master's student uses this subject first in the thesis: estimating a Cobb-Douglas production function from a farm survey, a growth-rate regression from district data, or a logit model of technology adoption. The practical tasks are to choose the functional form before looking at the results, check residual plots, compute VIFs, test for heteroscedasticity with White or Breusch-Pagan and report robust standard errors, and interpret coefficients in the units of the data (quintals per kg, elasticities, marginal effects on probability) rather than only stars. For the PhD entrance the student must also be able to state, for each assumption, what fails and what fix is standard, and to compute R-squared, F, DW and elasticities quickly by hand.
Five years into work in an agribusiness firm, bank, development agency or research institute, the analyst estimates demand for inputs, forecasts commodity prices, evaluates programmes and builds credit-scoring or adoption models. The recurring decisions are: whether a price series is stationary before using it in a model; whether to use ARIMA or a structural model for a three-month price forecast; how to handle the many zeros in input-use data (Tobit or double hurdle rather than OLS); whether an apparent programme effect is only self-selection; and how to explain to a manager that a coefficient with a wide confidence interval does not support a firm decision. Good practice is to keep a reproducible script, document every specification tried, and report out-of-sample forecast errors.
A senior economist or manager with 25 years of experience rarely runs regressions but decides which ones are believed. The skill is judging identification: is the price elasticity estimated from a demand curve that was actually identified; is the supply response estimate driven by a policy period; is a cointegration finding between two markets robust to structural breaks; did a consultant's high R-squared come from trending variables. They set standards for internal models (robust errors, out-of-sample validation, sensitivity to specification), commission impact evaluations with credible designs, and translate econometric uncertainty into risk ranges for procurement, pricing and policy decisions rather than point forecasts.
A professor or research leader with a PhD and 25 years of work teaches and examines this subject and supervises theses that use it. Their decisions concern research design: which source of exogenous variation identifies the effect of interest, whether panel fixed effects, instrumental variables, difference-in-differences or a regression discontinuity is credible in the setting, how to model spatial and temporal dependence in agricultural data, and how to handle non-stationary price panels. They review papers for journals, where the commonest faults are weak instruments, unit-root tests with no attention to breaks, marginal effects misread from logit coefficients and specification searching. They also update curricula so students learn modern causal-inference methods alongside the classical course.
This manual covers the classical first graduate course. It does not yet cover panel-data econometrics (fixed and random effects, the Hausman test between them, dynamic panels), modern causal-inference designs (difference-in-differences, regression discontinuity, synthetic control) beyond a mention, frontier production functions (stochastic frontier analysis), GARCH models of price volatility, multinomial, ordered and count-data models, propensity score matching and endogenous switching regression used in adoption studies, Granger causality beyond a paragraph, the Almon lag in detail, VAR impulse responses, threshold cointegration in detail, quantile regression or spatial econometrics. Critical values for t, F, Durbin-Watson and Dickey-Fuller tests must be read from statistical tables or software; only a few are quoted, and questions supply the values they need. Worked examples use invented illustrative numbers, not real estimates for any country or crop.
Original practice questions written to the syllabus and pattern of the exams named in each question's examTags. They are not past papers and do not reproduce any board's questions. Numbers in the numericals are invented for illustration and were computed by script; critical values are given in the question where a decision depends on them.
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