If the true data generating process is a random walk and you estimate an AR(1) by regressing y(t) on y(t-1), the OLS estimator is no longer asymptotically normally distributed and does not converge at the standard rate sqrt(T). Instead it converges to a limiting NON- NORMAL distribution that is tabulated by Dickey-Fuller Also, the OLS estimator converges at rate T and is SUPERCONSISTENT Principles of Econometrics, 3rd Edition
AS a consequence, the t-ratio statistics are not asymtptically normally distributed Instead, they have a NON-NORMAL limiting distribution, that was tabulated by Dickey- Fuller If you regress one unit root process on another the outcomes are meaningless (spurious) unless the processes are cointegrated Principles of Econometrics, 3rd Edition
12.3.2 Dickey-Fuller Test 2 (with constant but no trend)
12.3.3 Dickey-Fuller Test 3 (with constant and with trend)
First step: plot the time series of the original observations on the variable. If the series appears to be wandering or fluctuating around a sample average of zero, use test equation (12.5a). If the series appears to be wandering or fluctuating around a sample average which is non-zero, use test equation (12.5b). If the series appears to be wandering or fluctuating around a linear trend, use test equation (12.5c).
In each case you consider the t-statistic on the coefficient of y(t-1), called gamma, which is equal to zero if the true data generating process process is a unit root process That t-ratio is called “tau” and the critical values for tau are given in the table: Principles of Econometrics, 3rd Edition
An important extension of the Dickey-Fuller test allows for testing for unit root in AR(p) processes: AUGMENTED Dickey-Fuller. The unit root tests with the intercept excluded or trend included in AR(p) have the same critical values of tau as the AR(1)
A regression of a unit root process on another(s) makes sense ONLY if they are cointegrated i.e. satisfy a long run equilibrium and any departure from that equilibrium is eliminated by an Error Correction Model. If the processes are cointegrated, the residuals of the regression are stationary That regression represents the long run equilibrium Principles of Econometrics, 3rd Edition
To summarize: If variables are stationary, or I(1) and cointegrated, we can estimate a regression relationship between the levels of those variables without fear of encountering a spurious regression. If the variables are I(1) and not cointegrated, we need to estimate a relationship in first differences, with or without the constant term. If they are trend stationary, we can either de-trend the series first and then perform regression analysis with the stationary (de-trended) variables or, alternatively, estimate a regression relationship that includes a trend variable. The latter alternative is typically applied.
Slide 12-42 Principles of Econometrics, 3rd Edition Augmented Dickey-Fuller test Autoregressive process Cointegration Dickey-Fuller tests Mean reversion Order of integration Random walk process Random walk with drift Spurious regressions Stationary and nonstationary Stochastic process Stochastic trend Tau statistic Trend and difference stationary Unit root tests