Maddala Econometrics Solutions Pdf

[Alexandrie Gallup]

Thearticle provides a literature review on the topic of identification of supply and demand. In particular, it discusses the identification problem, that is the issue of having to solve for unique values of the parameters of the structural model from the values of the parameters of the reduced form of the model. We summarize several methodologies employed in the literature to solve this problem and gives practical examples. These solutions include, but are not limited to, using instrumental variables, adopting a recursive structure, holding demand constant, and imposing inequality constraints in order to restrict the domain of estimates. We also discuss on two major recent contributions in agricultural economics. The review will guide researchers in selecting the most suited approach to identify demand and supply.

Identification is a main issue in econometrics, the branch of economics that aims to answer to empirical questions based on economic models. Econometrics models are always based on assumptions, not always testable or falsifiable. In this framework, identification deals with the relationship between the assumptions of an econometric model and the possibility of answering or not, an empirical question using that model.

The present note aims at reviewing the status of art of identification in applied economics with particular emphasis to agricultural economics. The remainder of the note is as follows: section two summarizes the identification problem providing several definitions, the subsequent paragraph reviews the solutions that have been proposed in a century of research, and finally we conclude with final remarks1.

The area of identification studies the necessary and sufficient conditions to estimate (consistently) parameters of interest2. From a different perspective, the identification problem in econometrics is the issue of having to solve for unique values of the parameters of the structural model from the values of the parameters of the reduced form of the model (i.e. a single estimate of the structural parameters from the reduced form parameters for each structural equation, c f r. Maddala 1992)3. Therefore, if there are multiple solutions which make the reduced form coefficients compatible with the structural coefficients, the model is underidentified. Instead if there are no compatible solutions, the model is said to be overidentified. Finally, if a solution exists and is unique, the model is said to be just identified or exactly identified4.

Consider a linear model for the supply and demand: the former will be upward sloping with respect to price and the latter is expected to be downward sloping. We observe data on both the price (P) and the traded quantity (Q) of this good for several years. Unfortunately this information does not suffice to identify both demand and supply by using regression analysis on observations of Q and P. In fact it is impossible to estimate a downward slope and an upward slope with one linear regression line involving only two variables. Indeed, additional variables solve this issue and help to identify the individual relations. Put differently, by observing shifts in the demand (supply) curve, due to an exogenous variable, it is possible to identify the positive (negative) slope of the supply (demand) equation.

For instance, while we need demand shifters to estimate the slope of the supply, we need supply shifters to estimate the slope of the demand. More generally, we are able to identify the parameters of the equation (in our case the supply) not affected by the exogenous variable (Z). In order to identify both the supply and the demand equation, we would need both a variable (or shifter) Z entering the demand equation but not the supply equation (e.g. in agricultural economics it is common to use weather variables), and X entering the supply equation but not the demand equation (e.g. in agricultural economics a common approach is to introduce income as demand shifter). In other words, we need to introduce Z, a demand shifter (e.g. income) and X, a supply shifter (e.g. weather variable):

As previously described, the identification problem arises when we try to identify parameters using a reduced form. In the example of supply and demand, we may solve the problem by using an instrumental variable. Few points need to be recalled. More precisely, an instrument will be valid if the variable is correlated with the endogenous regressor and uncorrelated with the regression error.

Wright (1928) has pioneered the use of instrumental variables. He estimated they supply and demand for flaxseed and used prices of substituted goods as instrumental variables for demand, and yield per acre as instruments for supply. He averaged out the estimates obtained using different instruments. Current researches have shown that a more efficient way to rely on multiple instruments is to use a two-stage least squares (2SLS) procedure. The method is described below.

Frisch and Waugh (1933) have proposed another approach. They suggested to hold demand constant. Given that the observed quantity demanded differs from the true (or latent) demand, the approach consists of estimating the observed demand and correcting for the bias. We clarify with an example. Suppose that quantity is measured with error ε t , that is:

Another approach is to use an instrumental variables (IV) regression. In the case of a single equation, the Limited Information Maximum Likelihood method (LIML) is a valid alternative. The method has been proposed by Anderson and Rubin (1949), and has been popular until the introduction of the 2SLS by Theil (1965)5. The LIML consists in minimizing the residual sum of squares (RSS) to select the regressors. More precisely, the LIML minimize the ratio of RSS under the restricted and unrestricted model (Maddala 1992). The analogy with the 2SLS is very strong in that the latter minimize the difference of RSS under the restricted and unrestricted model. As a consequence, if the model is exactly identified the 2SLS and LIML provide identical estimates. Sargan (1958) has extended the IV approach to multiple instruments through the 2SLS method.

In a nutshell, the approach is as follows. In the first stage, each explanatory variable that is an endogenous covariate in the equation of interest is regressed on all of the exogenous variables in the model (including both exogenous covariates in the equation of interest and the excluded instruments). This first stage allows us to obtain the predicted values. In the second stage, the regression of interest is estimated as usual, except that in this stage each endogenous covariate is replaced with the predicted values from the first stage (Wooldridge 2010).

From an empirical point of view, it is worth recalling the pitfalls of instrumental variables approach. The 2SLS provides consistent, but not unbiased estimates, therefore researchers that use this approach should always aspire to use large datasets. Moreover, an instrumental variable correlated with omitted variables can lead to biased estimates that is much greater than the bias in ordinary least squares estimates. However, the bias is proportional to the degree of overidentification, hence using fewer instruments would reduce the bias. Moreover, it is wise to test for the validity of instruments. Many tests have been proposed and some are implement in common packages (see Berkowitz et al. 2012)6.

For the above mentioned approaches we have implicitly assumed to deal only with a single equation. Special attention needs the case in which we consider a simultaneous equation model. An efficient way to estimate a full system of equations is to use Generalized Method of Moments (GMM) estimation. Unfortunately, GMM is usually unfeasible, unless the system covariance matrix (Σ) is known. Alternative approaches consist in estimating the system by using a three stage least squares (3SLS) procedure, or by adopting a full information maximum likelihood (FIML) estimator. The former consists in estimating a 2SLS (or equation-by-equation) and then using the residuals to compute Σ. Using \(\widehat \Sigma \) the estimation of the third stage will be consistent. Alternatively a FIML estimator can be adopted. The estimator uses information about all the equations in the system, providing consistent estimates. Although asymptotically equivalent, the FIML is not equal to the continuously updated 3SLS estimator (unless the system is just-identified). Empirically, the 3SLS estimator is much easier to be computed than the FIML estimator (Davidson and MacKinnon 2004).

Alternative approaches have been proposed. Leamer (1981) has suggested to solve the identification problem by imposing inequality constraints in order to restrict the domain of estimates. His words are self-explanatory: when the regression of quantity on price yields a positive estimate, we may assume that this is an attenuated estimate of the supply curve and that the data contain no useful information about the demand curve.

Rigobon (2003) exploits the intuition in Wright (1928) suggesting to restrict the parametric space using the information provided by the heteroskedasticity in the data (e.g. due to crises, policy shifts, changes in collecting the data, etc.). He provides necessary and sufficient conditions for identification of a system of simultaneous equations. In particular, Rigobon suggests to use the second moments to increase the number of relations between the parameters in the reduced and structural forms. An appealing feature of his approach is that it only requires the existence of heteroskedasticity in that the direct modeling of the source of heteroskedasticity can be ignored for the identification purpose. The approach is as follows. First, Rigobon (2003) estimated a vector autoregressive model of interest rates (prices may be used for agricultural markets); second, he defined subsamples according to different volatility; finally he computed the covariances matrices that have been used in the GMM estimation of contemporaneous shocks. Although the intuition to use the variance of the shocks to reduce the bias in OLS estimates has to be attributed to Wright (1928), Rigobon (2003) generalized the intuition and provided the conditions to identify the system7.

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