24  Autocorrelation tests

24.1 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. $$\begin{array}{}\text{(24.1)}& x_t={\varphi }_1{x}_{t-1}+u_t\end{array}$$ 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.

24.2 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. $$\begin{array}{}\text{(24.2)}& x_t={\varphi }_1{x}_{t-1}+\cdots +{\varphi }_p{x}_{t-p}+u_t\end{array}$$ 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 24.2.

24.3 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.

IID assumption

Note that the assumption of the test is that the observations are IID. Therefore, the test do no apply in the case of heteroskedasticity.

24.3.1 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.  $${\text{LB}}_m=n(n+2)\sum _{k=1}^m\frac{{\hat{\rho }}_k^2}{n-k}\underset{H_0}{\stackrel{d}{⟶}}{\chi }^2(m)$$

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 $$\{\begin{array}{l}{\text{Q}}_m>{\chi }_{1-\alpha ,m}^2{\text{H}}_0\text{ rejected}\\ {\text{Q}}_m<{\chi }_{1-\alpha ,m}^2{\text{H}}_0\text{ not rejected}\end{array}$$ 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.