Autocorrelation tests

Author

Affiliation

Beniamino Sartini

University of Bologna

1 Durbin-Watson test

The aim of the Durbin-Watson test is to verify if a time series presents autocorrelation or not. Specifically, let’s consider a time series $X_t=(x_1,\dots ,x_i,\dots ,x_t)$, then evaluating an AR(1) model, i.e. $$\begin{array}{}\text{(1)}& x_t={\varphi }_1{x}_{t-1}+u_t\end{array}$$ we would like to verify if ${\varphi }_1$ is significantly different from zero. The test statistic, denoted as $\text{DW}$, is computed as: $$\text{DW}=\frac{\sum _{i=2}^t(x_i-x_{i-1}{)}^2}{\sum _{i=2}^t{x}_{i-1}^2}\approx 2(1-{\varphi }_1)$$ The null hypothesis $H_0$ is the absence of autocorrelation, i.e.  $$H_0:{\varphi }_1=0H_1:{\varphi }_1\ne 0$$ Under $H_0$ the Durbin-Watson statistic is approximated as $DW\approx 2(1-0)=2$. The test always generates a statistic between 0 and 4. However, there is not a known distribution for critical values. Hence to establish if we can reject or not $H_0$ when we have values very different from 2, we should look at the tables.

2 Breush-Godfrey

The Breush-Godfrey test is similar to Durbin-Watson, but it allows for multiple lags in the regression. In order to perform the test let’s fit an AR(p) model on the a time series $X_t=(x_1,\dots ,x_i,\dots ,x_t)$, i.e. $$\begin{array}{}\text{(2)}& x_t={\varphi }_1{x}_{t-1}+\cdots +{\varphi }_p{x}_{t-p}+u_t\end{array}$$ The null hypothesis $H_0$ is the absence of autocorrelation, i.e.  $$\begin{array}{rl}& H_0:{\varphi }_1=\cdots ={\varphi }_p=0\\ & H_1:{\varphi }_1\ne 0,\dots ,{\varphi }_p\ne 0\end{array}$$ The null hypothesis $H_0$ is tested looking at the F statistic that is distributed as a Fisher–Snedecor distribution, i.e $\text{F}\sim F_{p,n-p-1}$. Alternatively is is possible to use the $\text{LM}$ statistic, i.e. $\text{LM}=n{R}^2\sim \chi (p)$ where $R^2$ is the R squared of the regression in Equation 2.

3 Box–Pierce test

Let’s consider a sequence of $n$ IID observations, i.e. $u_t\sim \text{IID}(0,{\sigma }^2)$. Then, the autocorrelation for the $k$-lag can be estimated as: $${\hat{\rho }}_k=\mathbb{C}r\{u_t,u_{t-k}\}=\frac{\sum _{t=k}^n{u}_t{u}_{t-k}}{\sum _{t=k}^n{u}_t^2}\text{.}$$

Moreover, since ${\hat{\rho }}_k\sim N(0,\frac{1}{n})$, standardizing ${\hat{\rho }}_k$ one obtain
$$\sqrt{n}{\hat{\rho }}_k\sim N(0,1)⟹n{\hat{\rho }}_k^2\sim {\chi }_1^2\text{.}$$ It is possible to generalize the result considering $m$-auto correlations. In specific, let’s define a vector containing the first $m$ standardized auto-correlations. Due to the previous result it converges in distribution to a multivariate standard normal, i.e.  $$\sqrt{n}\left[\begin{array}{c}{\hat{\rho }}_1\\ ⋮\\ {\hat{\rho }}_k\\ ⋮\\ {\hat{\rho }}_m\end{array}\right]\underset{n\to \mathrm{\infty }}{\stackrel{d}{⟶}}\mathcal{N}({\mathbf{0}}_{m\times 0},{\mathbb{I}}_{m\times m})\text{.}$$ Remembering that the sum of the squares of $m$-normal random variable is distributed as a ${\chi }^2(m)$, one obtain the Box–Pierce test as $$B{P}_m=n\sum _{k=1}^m{\hat{\rho }}_k^2\underset{H_0}{\stackrel{d}{⟶}}{\chi }_m^2\text{,}$$ where the null hypothesis and the alternative are $$\begin{array}{rl}& H_0:{\rho }_1=\cdots ={\rho }_m=0\\ & H_1:{\rho }_1\ne 0,\dots ,{\rho }_p\ne 0\end{array}$$ Note that such test, also known as Portmanteau test, provide an asymptotic result valid only for large samples.

3.1 Ljung-Box test

Since the Box–Pierce test provide a consistent framework only for large samples, when dealing with a small samples it is preferable to use an alternative version, known as Ljung-box test, defined with a correction factor, i.e.  $$L{B}_m=n(n+2)\sum _{k=1}^m\frac{{\hat{\rho }}_k^2}{n-k}\underset{H_0}{\stackrel{d}{⟶}}{\chi }^2(m)$$

Independently from the statistic test used, i.e. $Q_m=B{P}_m$ or $Q_m=L{B}_m$, in general both are rejected when $$\{\begin{array}{l}{Q}_m>{\chi }_{1-\alpha ,m}^2{H}_0\text{ rejected}\\ Q_m<{\chi }_{1-\alpha ,m}^2{H}_0\text{ non rejected}\end{array}$$ where ${\chi }_{1-\alpha ,m}^2$ is the quantile with probability $1-\alpha$ of the ${\chi }_m^2$ distribution with $m$ degrees of freedom. If we reject $H_0$, the time series presents autocorrelation, otherwise if $H_0$ is non rejected we have no autocorrelation.