Disclaimer: This is a work in progress.

R code at the end.

Introduction

A fundamental utility in financial econometrics and stochastic processes is the martingale property, which implies that the best approximation of the future value of a time series or stochastic process, based on its historical values, is its present value. This property is critical in the efficient market hypothesis, risk-neutral pricing models. Testing whether a given time series satisfies the martingale hypothesis involves examining whether past values significantly predict future changes. This blog post outlines a formalized statistical test implemented in the esgtoolkit package, and leveraging multiple linear regression, F-statistics, and residual diagnostics to determine whether a time series follows a martingale process.

Martingale Hypothesis Test

Let \(X=\left\{X_{1}, X_{2}, \ldots, X_{n}\right\}\) be a time series, where each \(X_{t}\) represents a multivariate observation at time \(t\). We are interested in testing the martingale hypothesis, which posits that the future value \(X_{t+1}\) is unpredictable given the past values. That is:

\[\begin{equation*} \mathbb{E}\left[X_{n+1} \mid \sigma\left(X_{n}, X_{n-1}, \ldots, X_{1}\right)\right]=X_{n} \tag{1} \end{equation*}\]

Where \(\sigma\left(X_{n}, X_{n-1}, \ldots, X_{1}\right)\) is the sigma-algebra containing all the information about \(X^{\prime}\) ‘s past values.

1. Regression Model

One way to conduct such a test of the Martingale Hypothesis, is to adjust a multiple linear regression of the change in the series, \(\Delta X_{t+1}=X_{t+1}-X_{t}\), on the past values \(X_{t}, X_{t-1}, \ldots, X_{1}\), for all \(t>0\) :

\[\Delta X_{t+1} \approx \beta_{0}+\beta_{1} X_{t}+\beta_{2} X_{t-1}+\cdots+\beta_{p} X_{1}+\epsilon_{t+1}\]

where \(\epsilon_{t+1}\) are the (centered and homoskedastic) residuals of the regression, because, under the assumption that \(\hat{\beta}_{1}=\hat{\beta}_{2}=\cdots=\hat{\beta}_{p}=\) 0 , we’d have:

\[\begin{equation*} \mathbb{E}\left[X_{t+1}-X_{t} \mid \sigma\left(X_{t}, X_{t-1}, \ldots, X_{1}\right)\right]=0 \tag{2} \end{equation*}\]

That’s one of the simplest way to do it, and although we could think of other expressions of the conditional expectation, these would require more engineering.

2. F-statistic

The significance of the regression model is tested using the Fisher F-statistic. The null hypothesis \(H_{0}\) is that none of the past values \(X_{1}, X_{2}, \ldots, X_{t}\) significantly explains the change \(\Delta X_{t+1}\) :

\[H_{0}: \beta_{1}=\beta_{2}=\cdots=\beta_{p}=0\]

The F-statistic is computed as:

\[F=\frac{R^{2} / p}{\left(1-R^{2}\right) /(n-p-1)}\]

where \(R^{2}\) is the coefficient of determination, and \(p\) is the number of predictors (lags). A significantly large value of \(F\) would lead to reject \(H_{0}\).

3. Critical Value and p-value

The critical value for the Fisher-Snedecor statistic is obtained from the F-distribution with \(p\) and \(n-p-1\) degrees of freedom at a chosen significance level \(\alpha\) :

\[F_{\text {critical }}=F_{\alpha}(p, n-p-1)\]

The p -value of the F-statistic is computed as:

\[p \text {-value }=P\left(F \geq F_{\text {observed }} \mid H_{0} \text { is true }\right)\]

If the p-value is less than the chosen significance level \(\alpha\), we’ll reject the null hypothesis and conclude that the past values explain the change in the time series.

4. Stationarity of Residuals

To test the stationarity of the residuals \(\epsilon_{t+1}\), we perform the Augmented Dickey-Fuller (ADF) test. The null hypothesis \(H_{0}\) is that the residuals are non-stationary (i.e., they follow a random walk):

\[H_{0}: \text { Residuals are non-stationary }\]

The test statistic is the t-statistic of the lagged residuals, and the p-value indicates whether we can reject the null hypothesis. If the p -value is less than \(\alpha\), we reject the null hypothesis and conclude that the residuals are stationary.

5. Autocorrelation of Residuals

In addition to the F-test, and in order to ensure no autocorrelation remains in the residuals, we perform the Ljung-Box test. The null hypothesis \(H_{0}\) is that there is no autocorrelation in the residuals:

\[H_{0} \text { : No autocorrelation in residuals }\]

The Ljung-Box test statistic is computed as:

\[Q=n(n+2) \sum_{k=1}^{m} \frac{\hat{\rho}_{k}^{2}}{n-k}\]

where \(\hat{\rho}_{k}\) is the sample autocorrelation at lag k, and m is the maximum lag. The p -value is computed based on this statistic.

Examples

See: https://techtonique.github.io/ESGtoolkit/articles/martingale_test.html