Aggregate production function, factor shares and cointegration

When estimating an aggregate production function you fit your data to a selected functional form of the production function, derive the parameters and inference from there.

My question is, is there a reason for testing for cointegration (i.e. Engle & Granger method) of the production input variables' time series data? What is the extra information you could derive from, for example, VECM analysis of such cointegrated variables, apart from their long-term relationship?

1 Answer

Intuitively, you test for cointegration because if two variables are cointegrated, they represent only a "one dimensional" family of data points - even if you have a million data points from that sample, they will all fall close to the same subspace, and in general that will mean that you will have many values of parameters for your regression problem which will give a good fit (spurious regression).

Consider the linear regression case - say that you have $y = Ax + \varepsilon$ as your regression problem, where $x$ is a column vector, $A$ is a row vector and $y$ is a real number. Let's say further that the "structural" relationship we are looking for is $y = Bx$, where $B$ is also a row vector. Take an extreme case of cointegration - $x$ is $2$-dimensional, and we have a linear relationship between its coordinates, i.e $x_2 = c x_1$ for some constant $c$. In this case, any linear functional $A$ which is equal to $B$ when restricted to the subspace spanned by $(1, c)$ will give a perfect fit, and the regression won't be able to distinguish between them. (There's a one dimensional family of such functionals, i.e row vectors $A$.)

What VECM analysis does is to take advantage of deviations when the relationship is not perfect. In real econometric applications, you never have a relationship that's as clean as $x_2 = c x_1$ (or $y = Bx$) for your entire dataset - cointegration allows for transitory white noise in the linear combination $x_2 - c x_1$. If the variance of the white noise is relatively small, a standard regression will still return similar $R^2$ values for many different operators. Instead, what you do is look at first differences: a good fit should also fit these, i.e we should have

$$Ax_{t+1} - Ax_t \approx y_{t+1} - y_t$$

where the subscripts denote the time of observation. Running the regression this way allows you to capture the white noise in the cointegration process and use it to your advantage - it's a way of capturing the information which a naive linear regression is going to miss.