public double VarSEbyJ(int j)
{
var new_y = DenseVector.OfVector(x.Column(j));
var new_x = DenseMatrix.OfMatrix(x.RemoveColumn(j));
var helpRegression = new MultipleRegressionModel();
helpRegression.Fit(new_y, new_x, !intercept); // here x already has a constant column, so the regression model should not add another
return(Variance(errors) / (Variance(new_y) * (1 - helpRegression.r_squared)));
}
// Regress one timeseries on the other, test if the residuals of this regression satisfy requirement of stationarity by performing the Augmented Dickey Fuller Test on them and returning its p-Value
public float EngelGrangerRegression(double[] values1, double[] values2, int lag)
{
if (values1.Length != values2.Length)
{
throw new ArgumentException($"EngelGranger test requires bot inputs to be of same lenght, the provided inputs are of length {values1.Length} and {values2.Length}");
}
// The Engel-Granger Two Step Test for Cointegration, requires a method (like ADF) to test time series data for stationarity
// simple regression of values1 = beta_0 + beta_1 * values2 + error
var regModel = new MultipleRegressionModel();
var y = DenseVector.OfArray(values1);
var x = DenseMatrix.OfColumnArrays(values2);
regModel.Fit(y, x);
float[] u = regModel.errors.Select(i => (float)i).ToArray();
// return ADF for the residuals
return(ADF(u, lag));
}
// Perform the Augmented Dickey Fuller Test on a timeseries with a given number of lags
public float ADF(float[] values, int lag = 1)
{
// To keep the regression feasable in terms of the proportion of features and observations, the lag will be restricted to the squareroot of all observations
// This relation was arbitrarly choosen and could be changed -> however a decreasing growth rate for the lag in the number of observations, is desirable for run time efficiency and computational precision
if (lag > Math.Sqrt(values.Length))
{
lag = Convert.ToInt32(Math.Sqrt(values.Length));
} // A warning could be added
// The reasoning of the following code follows the ADF as described here: https://nwfsc-timeseries.github.io/atsa-labs/sec-boxjenkins-aug-dickey-fuller.html
// The ADF performs a regresssion where the Dependent variable y_delta is regressed on the lag one of y_t and multiple lags of the differenced timeseries
// First calculate the set of dependent variables
double[] getYDifferences()
{
double[] results = new double[values.Length - 1];
for (int i = 0; i < results.Length; i++)
{
results[i] = (double)values[i + 1] - values[i];
}
return(results);
};
double[] YDifferences = getYDifferences();
// Because the ADF requires YDifferences to be regressed on its own lags (p), the number dependent variables for the observation are actually YDifferences.Lenght - lag
int length = YDifferences.Length - lag;
// alpha will be generated by allowing the regression to have an intercept, the other parameters are defined in the lines below
// beta is defined as being the coefficient of t, thus the regression will require a count variable for the observatins
int[] t_values = Enumerable.Range(0, length).ToArray();
// The regression requires the last <length> observations of y_t with lag one, this is the parameter whose coefficient will be used to create the test statistic
float[] y_1 = new float[length];
Array.Copy(values, lag, y_1, 0, length); // lag does not need to be corrected by -1 as the YDifferences are calculated looking forward
// define a function to get the differences in yDifferences with the desired shift -> shift 0 yields the dependent variable
double[] YDifferencesWithLag(int shift)
{
double[] yDiff = new double[length];
Array.Copy(YDifferences, lag - shift, yDiff, 0, length);
return(yDiff);
}
// Y_t - Y_{t-1}-> the dependent variable of the regression
var y = DenseVector.OfArray(YDifferencesWithLag(0));
// define a matrix with the lagged YDifferences, as features, a constant to facilitate an intercept in the regresion is added by the regressionmodel
var x_values = new Vector <double> [lag + 2]; // plus 2 for t_values and y_1
x_values[0] = new DenseVector(t_values.Select(i => Convert.ToDouble(i)).ToArray());
x_values[1] = new DenseVector(y_1.Select(i => (double)i).ToArray());
// Fill the columns of the matrix with the lagged timeseries values (lag is increasing -> i)
for (int i = 0; i < lag; i++)
{
float[] ydiff_t_i = new float[length];
Array.Copy(values, lag - (i + 1), ydiff_t_i, 0, length);
x_values[i + 2] = new DenseVector(Array.ConvertAll(ydiff_t_i, v => (double)v));
}
var x = DenseMatrix.OfColumns(x_values);
// Perform a regression
var rm = new MultipleRegressionModel();
rm.Fit(y, x);
// return the p-Value for the Intercept
float testStatistic = (float)(((float)rm.betas.ToArray()[2]) / Math.Sqrt(rm.VarSEbyJ(2))); // cf. for ADF statistic with mutliple regressors (not shown in the link above) https://en.wikipedia.org/wiki/Augmented_Dickey%E2%80%93Fuller_test
return((float)(StudentT.CDF(0, 1, length - x.ColumnCount, testStatistic))); // The test is a onesided, testing for t-values beyond the critical threshold -> a sufficiently low CDF value here implies stationarity
}
