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.
Maddala (1977) pointed it is very difficult to have such kind of a variable, and econometrics textbooks do not provide clear guidelines. Angrist and Krueger (2001), p. 73, argue that “good instruments often come from detailed knowledge of the economic mechanism and institutions determining the regressor of interest”. For example, a valid instrument shifts only one “curve” (e.g. supply, but not demand). In agricultural markets, the instrument may be rainfall or weather shocks.
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.
First we provide a chronological review of the solutions have been proposed to solve the identification problem.
A simple, probably too naive, solution is to ignore the problem. Indeed this solution is not lacking of a theoretical justification. As Wright (1929) pointed in JASA, ignoring the issue is a valid solution if “it may be assumed that the dynamic forces will continue to operate thereafter in the same manner as they have been operating during that period”.
Another solution is to adopt a recursive structure:
$$\textrm{(i)} {p}_{t}=\beta_{1}{q}_{t}+u^{D} \textrm{(ii)} {q}_{t}=\beta_{2} {p}_{t-1}+u^{S}. $$
In this formulation p
t−1 is exogenous in the supply equation, u
S is uncorrelated with u
D (therefore there are no common shifters), and q
t
is exogenous in the demand equation with p
t
on the left hand side.
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:
$$\textrm{(true demand)}{q}_{t}^{*}=\beta {p}_{t}+\gamma {W}_{t} \textrm{(observed quantity)}{q}_{t}={q}_{t}^{*}+\epsilon{\!~\!}_{t} $$
where W
t
represents all determinants of demand and ε
t
is pure independent measurement error. Solving for observed demand:
$$\textrm{(observed demand)}\ {q}_{t}=\beta {p}_{t}+\gamma {W}_{t}+\epsilon_{t} $$
where E(p
t
ε
t
)=0. The approach suggested by Frisch and Waugh (1933) is to adjust for the bias, given the “known” γ and W
t
. In this case, as they prove, OLS estimates are consistent.
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).
Empirically, the 2SLS is performed as follows. Let y be the dependent variable, x
1,…,x
k−1 the explanatory variables, x
k
the endogenous regressor, z
1,…,z
M
the set of instruments.
(I) First stage: compute \(\hat {x}_{k}\) regressing x
k
on regressors and instruments.
$$ \ x_{k}=\alpha+{\sum}_{i=1}^{k-1}\beta_{i}x_{i}+{\sum}_{j=1}^{N}\gamma_{j}zj $$
((1))
(II) Second stage: estimate the model replacing x
k
with \(\hat {x}_{k}\).
$$ y=\alpha+{\sum}_{i=1}^{k-1}\beta_{i}x_{i}+\beta_{k}\hat{x}_{k} $$
((2))
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.
If the estimate is negative, the number may be treated as an attenuated estimate of the demand slope, and we may assert that the data contain no useful information about the supply curve” (Leamer 1981), p. 321. Thurman and Wohlgenant (1989) provide an empirical application of Leamer’s method in agricultural markets for the estimation of demand, whereas Renuka and Kalirajan (2002) applied the method to the demand for services. More recently, Garnache and Mérel (2015) use a mathematical programming framework, and a set of constraints to identify crop supply elasticities.
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.
Roberts and Schlenker (2013) have revisited the problem of identification of supply and demand for agricultural commodities. The authors use theory of storage to derive the following empirical model:
$$\begin{array}{@{}rcl@{}} &&\textrm{(Supply)} log(s_{t})=\alpha_{s}+\beta_{s}log(E[p_{t}|\Omega_{t-1}])+\gamma_{s}w_{t}+f(t)+u_{t} \end{array} $$
((3))
$$\begin{array}{@{}rcl@{}} &&\textrm{(Demand)} log(c_{t})=\alpha_{d}+\beta_{d}log(p_{t})+g(t)+v_{t} \end{array} $$
((4))
and c
t
=s
t
−z
t
(consumption, c
t
, is the difference of supply, s
t
, and storage, z
t
), α
s
and α
d
are intercepts for supply and demand, the Ω is the information set, w
t
stands for the random weather-induced yield shocks, f(t) and g(t) capture time trends in supply and demand, u
t
and v
t
are the error terms. The rationale for (24) and (25) is that weather-induced shocks (current and lagged) are expected to shift only the supply curve, and to leave the demand unchanged. The model is solved in two stages. The first stage consists in estimating l
o
g(p
t
) and l
o
g(E[p
t
|Ω
t−1]). The authors suggest to use a distributed lag model of yield shocks and a polynomial time trend. The reduced forms are as follows:
$$ log(p_{t})=\pi_{d0}+{\sum}_{k=1}^{K-1}\mu_{d}w_{t-k}+f(t)+\epsilon_{dt} $$
((5))
$$ log(E[p_{t}|\Omega_{t-1}])=\pi_{s0}+{\sum}_{k=1}^{K}\mu_{s}w_{t-k}+f(t)+\epsilon_{st} $$
((6))
where f(t) and g(t) represent the polynomial time trend functions, ε
dt
and ε
st
are the error terms. In the second stage the lagged yield shocks are used as instruments. In particular the supply is estimated as follows:
$$ log(s_{t})=\alpha_{s}+\beta_{s}\widehat{log(E[p_{t}|\Omega_{t-1}])}+\lambda_{s0}\gamma_{s0}w_{t}+f(t)+u_{t} $$
((7))
and demand is obtained as follows:
$$ \textrm{(Demand)}\,\,\, log(c_{t})=\alpha_{d}+\beta_{d}\widehat{log(p_{t})}+g(t)+v_{t} $$
((8))
The novelty of this approach is that Roberts and Schlenker (2013) have considered simultaneously four commodities that are substitutes in supply and demand, and have instrumented supply by using weather shocks8.
More recently, Steinwender (2014) has proposed a novel approach to identify the demand equation. Starting from a simple two markets model, and allowing for trade and storage, the identification problem may occur if the unobserved demand shocks are positively correlated with change in stock and exports. Put differently, because quantities and prices are determined contemporaneously, we need a valid instrument to estimate them correctly. Steinwender (2014) proposed to use the fact that exports (which take k periods to reach destination) are predetermined at destination as instruments to identify demand shocks. The demand equation tales the following form:
$$ \textrm{(Demand)}\,\,\, p_{t+k}=\alpha_{d}+\beta_{d} (x_{t} - \Delta s_{t+k})+v_{t+k} $$
((9))
where the price p
t+k
is function of exports (x
t
) that reach location at time k, and the change in stock (Δ
s
t+k
) at time t+k. The approach is interesting in that it does not require other data than exports, stock changes, and prices. A drawback is that stock data are usually not available at the same time frequency as trade and price data: price and trade data are usually at monthly, weekly, and also daily frequency, whereas stock data are rarely available at such a high frequency.
