22 Unit Root Process (contd)
22.1 Univariate case
We now have all the ingredient to further analyse the unit root process
\[ y_t = \phi y_{t-1} + \epsilon_t, \]
where \(\phi=1\), and its OLS estimator
\[ T(\hat\phi-1) = \frac{T^{-1}\sum_t y_{t-1}\epsilon_t}{T^{-2}\sum_t y_{t-1}^2}. \]
We have shown that
\[ \begin{aligned} T^{-1}\sum_t y_{t-1}\epsilon_t &\to \sigma^2\int_0^1 W dW =\frac{\sigma^2}{2}(W^2(1)-1)\\ T^{-2}\sum_t y_{t-1}^2 &\to \sigma^2\int_0^1 W^2 ds \end{aligned} \]
Therefore,
\[ T(\hat\phi -1) \to \frac{\int_0^1 W dW}{\int_0^1 W^2 ds}. \]
\(\int W dW\) is centered around \(0\), meaning \(\hat\phi\) is consistent for large samples. But it is biased in small smaples. Moreover, the distribution is not Gaussian, rending all conventional \(t\)-test or \(F\)-test meaningless. We contrast the properties of stationary processes and unit root processes below.
| Stationary | Unit Root | |
|---|---|---|
| Model | \(y_t =\phi y_{t-1} +\epsilon_t\) | \(y_t = y_{t-1}+\epsilon_t\) |
| Asymptotic distribution of \(\hat\phi\) | \(\sqrt{T}(\hat\phi-\phi)\to N(0,1-\phi^2)\) | \(\sqrt{T}(\hat\phi-1)\to\frac{\int WdW}{\int W^2 dt}\) |
| Asymptotic distribution of \(t\)-statistics | \(t \to N(0,1)\) | \(t \to \frac{\int WdW}{\sqrt{\int W^2dt}}\) |
22.2 Spurious regression
We now dive deeper into the nature of spurious regression problem presented at the beginning of the chapter. We formulate the problem as below. Suppose
\[ y_{t} = \alpha + \beta x_{t} + u_t, \]
where \(y_{t}\) and \(x_{t}\) are unit root processes and there does not exist \((\alpha, \beta)\) such that the residual \(u_t\) is stationary. In this case, OLS is likely to produce spurious result: even if \(y_{t}\) is completely unrelated to \(x_t\), the estimated value of \(\hat\beta\) is likely to appear to be statistically significantly different from zero.
Spurious regression happens when
- Dependent/independent variables are non-stationary;
- The residual is non-stationary for all possible values of the coefficient vector.
To understand why this happens, consider the OLS estimator:
\[ \hat b=\begin{bmatrix}\hat\alpha \\ \hat\beta\end{bmatrix} = \begin{bmatrix}T & \sum x_t \\ \sum x_t & \sum x_t^2 \end{bmatrix}^{-1} \begin{bmatrix}\sum y_t \\ \sum x_ty_t\end{bmatrix} \]
To account for different convergent speed, similar to the trend-stationary case, we multiply the estimators by a matrix,
\[ \begin{aligned} \begin{bmatrix}\sqrt{T}^{-1} & \\ & 1\end{bmatrix} \begin{bmatrix}\hat\alpha \\ \hat\beta\end{bmatrix} &= \begin{bmatrix}\sqrt{T}^{-1} & \\ & 1\end{bmatrix} \begin{bmatrix}T & \sum x_t \\ \sum x_t & \sum x_t^2 \end{bmatrix}^{-1} \begin{bmatrix}\sqrt{T}^{-1} & \\ & 1\end{bmatrix}^{-1} \begin{bmatrix}\sqrt{T}^{-1} & \\ & 1\end{bmatrix} \begin{bmatrix}\sum y_t \\ \sum x_ty_t\end{bmatrix} \\ &= \begin{bmatrix}1 & T^{-3/2}\sum x_t \\ T^{-3/2}\sum x_t & T^{-2}\sum x_t^2 \end{bmatrix}^{-1} \begin{bmatrix}\sum T^{-3/2}y_t \\ T^{-2}\sum x_ty_t\end{bmatrix} \\ &\to \begin{bmatrix}1 & \int W_X dt \\ \int W_X dt & \int W_X^2 dt\end{bmatrix}^{-1} \begin{bmatrix}\int W_Y dt \\ \int W_X W_Y dt\end{bmatrix} \end{aligned} \]
This means, \(\hat\alpha\) actually diverges. Because it needs to be divided by \(\sqrt T\) to be able to converge to a stable distribution, rather than being multiplied by a stabilizing factor. \(\hat\beta\) converges, but it is not consistent. If there is no \(\alpha\), we would have
\[\hat\beta = \frac{\sum x_ty_t}{\sum x_t^2}\to\frac{\int W_XW_Y dt}{\int W_X^2 dt},\]
which is inconsistent. So we won’t get zero even in very large samples.
The OLS estimate of the variance of \(u_t\) also diverges. It needs to be divided by \(T\) to converge:
\[ \begin{aligned} \frac{1}{T}\hat\sigma^2 &= \frac{1}{T^2}\sum(y_t-\hat\beta x_t)^2 \\ &= \frac{1}{T^2}\sum y_t^2 - 2\hat\beta\frac{1}{T^2}\sum x_ty_t + \hat\beta^2\frac{1}{T^2}\sum x_t^2 \\ &\to \int W_Y^2 dt - 2\hat\beta\int W_X W_Y dt + \hat\beta^2 \int W_X^2 dt \\ &\to \int W_Y^2 dt - \frac{(\int W_X W_Y dt)^2}{\int W_X^2 dt}. \end{aligned} \]
The \(t\) or \(F\) statistics also diverge. \(t\)-stat has to be divided by \(\sqrt T\) to converge; \(F\)-stat needs to be divided by \(T\) to converge.
\[ t = \frac{\hat\beta}{\hat\sigma}\sqrt{\sum x_t^2} = \sqrt{T}\frac{\hat\beta}{\sqrt{T^{-1}\hat\sigma^2}}\sqrt{T^{-2}\sum x_t^2} \to \sqrt{T}\cdot C \]
Thus, as sample size \(T\) grows, \(t\)-test will appear very large and significant, despite \(y_t\) and \(x_t\) are completely independent.
22.3 Cures for spurious regression
Include lagged values of both dependent and independent variables in the regression: \[ y_t = \alpha + \phi y_{t-1} + \beta x_t + \gamma x_{t-1} + u_t \]Now there exists a coefficient vector \([\phi,\beta,\gamma]=[1,0,0]\) such that \(u_t\) is \(I(0)\) stationary. In this case, OLS yields consistent estimates for all the coefficients. \(t\)-test converges to Gaussian, though \(F\)-test of joint hypotheses has non-standard asymptotic distribution. We will come back to this point later.
Difference the data to stationary: \[ \Delta y_t = \alpha + \beta\Delta x_t + u_t \] Because \(\Delta y_t\) and \(\Delta x_t\) are all \(I(0)\). Standard OLS is valid.
Estimate with Cochrane-Orcutt adjustment for first-order correlations in the residuals. This method is asymptotically equivalent to the second method.
22.4 Summary
- The OLS estimator for unit root coefficient converges to non-standard distributions involving Brownian motions. Thus, standard statistical inferences are meaningless.
- Regressing unit root processes lead to spurious results, because the diverging behavior of \(t\)-stats makes artificially significant values.
- Include lagged values or difference the data to stationary when working with non-stationary time series.