How to run a Hausman Specification Test in R

Consider the following structural equations: \begin{aligned} M_{t}^{d} &= \beta_{0} + \beta_{1} Y_{t} + \beta_{2} R_{t} + \beta_{3} P_{t} + u_{1t}\\ M_{t}^{s} &= \alpha_{0} + \alpha_{1} Y_{t} + u_{2t} \end{aligned} where $M$ = money, $Y$ = income, $R$ = rate of interest, $P$ = price, and the $u$'s are error terms. Assume that $R$ and $P$ are exogenous and $M$ and $Y$ are endogenous.

Let's begin by deriving the reduced form equations for the model.

By the equilibrium condition, $M_{t}^{d} = M_{t}^{s}$, we have: \begin{aligned} \beta_{0} + \beta_{1} Y_{t} + \beta_{2} R_{t} + \beta_{3} P_{t} + u_{1t} &= \alpha_{0} + \alpha_{1} Y_{t} + u_{2t}\\ \beta_{1} Y_{t} - \alpha_{1} Y_{t} & = \alpha_{0} - \beta_{0} - \beta_{2} R_{t} - \beta_{3} P_{t} + u_{2t} - u_{1t}\\ Y_{t} (\beta_{1} - \alpha_{1}) &= \alpha_{0} - \beta_{0} - \beta_{2} R_{t} - \beta_{3} P_{t} + u_{2t} - u_{1t}\\ Y_{t} &= \frac{\alpha_{0} - \beta_{0}}{\beta_{1} - \alpha_{1}} - \frac{ \beta_{2} R_{t}}{\beta_{1} - \alpha_{1}} - \frac{\beta_{3} P_{t}}{\beta_{1} - \alpha_{1}} + \frac{u_{2t} - u_{1t}}{\beta_{1} - \alpha_{1}}\\ Y_{t} &= \frac{\alpha_{0} - \beta_{0}}{\beta_{1} - \alpha_{1}} - \frac{ \beta_{2}}{\beta_{1} - \alpha_{1}}R_{t} - \frac{\beta_{3} }{\beta_{1} - \alpha_{1}}P_{t} + \frac{u_{2t} - u_{1t}}{\beta_{1} - \alpha_{1}}\\ Y_{t} &= \Pi_{0} + \Pi_{1}R_{t} + \Pi_{2}P_{t} + w_{t} \end{aligned} where $\Pi_{0} = \frac{\alpha_{0} - \beta_{0}}{\beta_{1} - \alpha_{1}}$ , $\Pi_{1} = - \frac{ \beta_{2}}{\beta_{1} - \alpha_{1}}$, $\Pi_{2} = - \frac{\beta_{3} }{\beta_{1} - \alpha_{1}}$ , and $w_{t} = \frac{u_{2t} - u_{1t}}{\beta_{1} - \alpha_{1}}$.

Substituting $Y_{t} = \Pi_{0} + \Pi_{1}R_{t} + \Pi_{2}P_{t} + w_{t}$ into either the supply function or the demand function, we can solve for the second reduced form equation which is: \[M_{t} = \Pi_{3} + \Pi_{4}R_{t} + \Pi_{5}P_{t} + v_{t}.\] Now let's find out if the structural equations are identified or not identified.

A structural equation can be said to be identified or unidentified. Furthermore, identified equations can be classified as just-identified, overidentified, or, underidentified. To determine the identification of structural equations we can use the order and rank conditions of identification. The order condition of identifiability is defined (in Gujarati's Basic Econometrics) as follows:

In a model of $M$ simultaneous equations, in order for an equation to be identified, it must exclude at least $M - 1$ variables (endogenous as well as predetermined) appearing in the model. If it excludes exactly $M - 1$ variables, the equation is just identified. If it excludes more than $M - 1$ variables, it is overidentified.

There are two endogenous variables in the model, namely, $Y_{t}$ and $M_{t}$, so in this case $M = 2$. This means that the demand function should omit at least $2 - 1 = 1$ variable in order to be identified. Since the demand function excludes $0$ variables from the model, it is unidentified. The supply function, however, omits $2$ variables, and is thus classified as overidentified.

With the preliminaries out of the way, we can now focus on the Hausman specification test.

To carry out the test on the supply function, there are two steps.

The first step is to regress $Y_{t}$ on $R_{t}$ and $P_{t}$. So, estimate by OLS the reduced form equation: $Y_{t} = \Pi_{0} + \Pi_{1}R_{t} + \Pi_{2}P_{t} + w_{t}$. Then, obtain the residuals, denoted $\hat{w_{t}}$, from this regression.

The second step is to regress $M_{t}$ on $Y_{t}$ and $\hat{w_{t}}$, and then perform a t-test on the coefficient of $\hat{w_{t}}$. The null hypothesis is that there is no simultaneity.

OK. Let's do this in R using some actual data.

Here is an R script that I've written to show how it can be done. It happens to be rather easy.

The dataset that I've used comes from an empirical exercise at the end of Chapter 19 in Gujarati's Basic Econometrics. The dataset is downloadable as a zip file from the book's website and the specific .txt file for the data is named Table 19.4.txt. You can go fetch it yourself or to save you the hassle, here is the data.

Before you run my R script you will need to change "/home/name/Desktop/hausmanData.txt" in the scan() function to the location of the dataset saved on your computer.