Diebold-Mariano Test

The test originally proposed by Diebold and Mariano (1995) considers a sample path of loss differentials \(\{d_{t}\}^{T}_{t=1}\). In the case of a squared loss function, we have \(d_{t}=e^2_{t}-\breve{e}^2_{t}\). Under the assumption that the loss differential is a covariance stationary series, the sample average, \(\bar{d}\), converges asymptotically to a normal distribution:

In particular, they proposed to test the null hypothesis that the forecast errors coming from the two forecasts bring about the same loss:
\(E[e^2_{t}-\breve{e}^2_{t}]=0\) against the two-sided alternative. Thus, the resulting p-values represent the probability of obtaining the realized forecast error differential or a more extreme one in a new experiment if the null hypothesis was actually true.

DM test-statistic

The test-statistic that will be used to calculate our p-values is computed as follows:

\begin{equation} DM=\dfrac{\bar{d}}{\sqrt{\dfrac{2\pi\hat{f}_{d}(0)}{T}}} \label{DMTEST} \end{equation}

where \(2\pi\hat{f}_{d}(0)\) is a consistent estimate. Consider \(2\pi\hat{f}_{d}(0)=\sum^{(T-1)}_{\tau=-(T-1)}w_{\tau}\gamma_{d}(\tau)\) , where \(\gamma_{d}(\tau)=\dfrac{1}{T}\sum^{T}_{t=|\tau|+1}(d_{t}-\bar{d})(d_{t-|\tau|}-\bar{d})\) . Under the assumption that \(\gamma_{d}(\tau)=0\) for \(\tau\geq h\) , we can use a rectangular lag window estimator by setting \(w_{\tau}=0\) for \(\tau\geq h\) . Another option is to use the Heteroschedasticity and Autocorrelation Consistent (HAC) estimator proposed by Newey and West (1987). In this case, the weights could be given by a triangular window, \(w_{\tau}=1-\dfrac{\tau}{h}\) for \(\tau<h\) . In this case, however, the consistency property only remains valid when the truncation lag $h$ or bandwidth is a function of the sample size \(T\).

The idea is to test the statistical significance of the regression of \(e^2_{t}-\breve{e}^2_{t}\) on an intercept. In order to determine the statistical significance of the intercept, its associated standard errors need to take into account the autocorrelation patterns of the regression error, which are considered in the denominator of equation (\ref{DMTEST}).

Small samples

The small sample sizes that are typical in real-time forecasting applications lead to an over-rejection of the null hypothesis under standard asymptotics, so we follow the fixed-smoothing asymptotics proposed by Coroneo and Iacone (2015). The idea is to use the finite sample distributions of Kiefer and Vogelsang (2005). As a result, the distribution of the test statistic (\ref{DMTEST}) will depend on kernel (triangular) and the bandwidth chosen, which is set by default equal to $T^{0.5}$. The results can be very different than those resulting from the traditional asymptotic theory, where the test statistic would have the same distribution under the null independently of the kernel and the bandwidth used.

JDemetra+ implementation

Class structure

JDemetra+ exploits the same unified framework to conduct all forecasting accuracy tests.

• The class AccuracyTests contains all methods required to perform the tests. All calculations are equivalent independently of the kind of test because the calculation of the loss function $ d_{t} $ is defined using abstraction, which is one of the features of the Java programming language. As a result, this class is extended by several classes that incorportate a precise implementation of the method to calculate the loss function: BiasTest, EfficiencyTest, DieboldMarianoTest and EncompassingTest. The constructor of each one of these classes can generate the tests when either the forecasts or the forecast errors are given as an input. Another input required is a boolean (AsymptoticsType) specifying whether standard asymptotics or fixed-smoothing asymptotics.

• All the tests contained in the class AccuracyTests will be constructed using the class GlobalForecastingEvaluation, which contains the various tests as objects. This is illustrated in the following example.

• The class ForecastEvaluation contains methods to quantify errors: Root Mean Squared Errors (RMSE), relative RMSE, Mean Absolute Errors (MAE), etc… Those statistics could be reported along with the test results.

A simple example

Suppose we want evaluate the forecasts of a model and compare them with those of a benchmark. The following points explain all the steps followed in the code below to run all the tests:

• First we need to initialize an array of time series TsData[] that includes the two competing forecast (i.e. benchmark vs model) and the target. Next, we initialize the p-value corresponding to the test, and the RMSE, which will be calculated at the end.
• Second, we initialize the eval object of the class GlobalForecastingEvaluation, which will contain all test results. The inputs needed to run the tests are three time series: our model’s forecasts, those of the benchmark, and the actual data, which is refered to as the target. We also need to specify the kind of distribution of the various test statistics under the null, which is given by a normal distribution when AccuracyTests.AsymptoticsType.STANDARD_FIXED_B is used. By choosing the option AccuracyTests.AsymptoticsType.HAR_FIXED_B, the distribution tabulated by Kiefer and Vogelsang (2005) is used.
• Since eval belongs to the class GlobalForecastingEvaluation, which contains all tests, the instruction eval.getDieboldMarianoTest() will trigger the necessary calculations. In this example, we will be asking for the pvalue of the Diebold-Mariano test.

• For each type of test, the bandwidth used to estimate the variance needs to be specified. Otherwise, the default value will be used (\(T^{1/2}\)). The relevant statistics for each test as well as the pvalues are obtained with a simple get command. Notice that getPValue(twoSided) uses the logical argument true in order to get the p-values of the two-sided test.

• Finally, the feval object of the class ForecastEvaluation is initialized in order to calculate the RMSE.

	public void example() {  
		TsData[] series = {benchmark, model, target};

		double rmse = new double ;
		double dmPval = new double ;
		boolean twoSided = true;
			
		int bandwith = (int) Math.pow(series.getObsCount(), 1.0 / 2.0);
			
		GlobalForecastingEvaluation eval = new GlobalForecastingEvaluation(model, benchmark, target,
		AccuracyTests.AsymptoticsType.HAR_FIXED_B);
		eval.getDieboldMarianoTest().setBandwith(bandwith);    
		dmPval = eval.getDieboldMarianoTest().getPValue(twoSided); 
			
		ForecastEvaluation feval = new ForecastEvaluation(model, benchmark, target);
		rmse = feval.calcRMSE();   
	}