Statistical Testing

	Attachment	Size
	TUTORIAL- STAT TESTS.xls

Earlier, we computed a few statistical parameters for the underlying daily returns distribution using sample data, but what can we infer about the actual values? Are the values we computed biased estimates of true values? Are these values, given the sample size, significantly different from zero or some other given value?

To examine the statistical significance of the computed values, we construct a series of hypotheses and compute the probably of accepting/rejecting the null hypothesisⁱ.

Basic Statistical Tests

Let's look at the sample mean above, -0.03%, and form a hypothesis that it is not significantly different from zero.

$H_o:\mu=0$

$H_1:\mu\neq 0$

Where:

$H_o$ : Null hypothesis

$H_1$ : Alternative hypothesis

$\mu$ : Population mean

To accept or reject the hypothesis:

We set the probability of incorrectly rejecting the null hypothesis as the significance level ( $\alpha$ )
Assuming the null hypothesis is true, compute the conditional probability of observing at least this sample mean (p-value)

In summary,

Accept $H_o$ if the p-value is higher than the significance level ( $\alpha$ ).
Reject $H_o$ otherwise.

Statistical test output

Advanced Statistical Tests

Computing the descriptive statistics gave us some idea about the distribution of the log returns; it has a small mean not significantly different from zero, a standard deviation of 1.88%, positive skew and fat tails on both sides.

Next, we ask the following questions:

Q: Do the daily log returns exhibit serial correlations or white noise characteristics?
A: White-Noise Test
Q: Do the daily log returns follow a normal distribution?
A: Normality Test
Q: Do the squared daily log returns exhibit serial correlations or white noise characteristics?
A: ARCHⁱ Effect Test

White-Noise Test

In this test, we examine whether the p autocorrelation factors are all equal to zero. The null hypothesis is rejected if at least one autocorrelation factor is statistically different from zero.

$H_{o}: \rho_{1}=\rho_{2}=...=\rho_{p-1}=\rho_{p}=0$

$H_{1}: \exists\rho_{k}\neq 0; k \in [1, p]$

Where

$H_o$ : Null hypothesis (White Noise)

$H_1$ : Alternative hypothesis (No White Noise)

$\rho_k$ : Autocorrelation function for k lags

$p$ : Upper lag limit of the test

The upper lag limit can either be set by us or left for the function to pick a proper limit. Practitioners use ln(t) as a proper value for p. In our example we did not specify p, so the function calculated a p-value for us.

White-noise Test Output

Normality Test

Similar to what we did with the sample mean test, we start by formulating a hypothesis, then we set the desired significance level, and finally we compute a p-value for our sample.

$H_o:r_t \sim N(.)$

$H_1:r_t \neq N(.)$

Where

$H_o$ : Null hypothesis (Normally Distributed)

$H_1$ : Alternative hypothesis (Not Normally Distributed)

$N(.)$ : Normal distribution

There are a number of normality tests (e.g. Jarque-Bera, Shapiro-Wilk, etc.), and their computed p-values differ slightly, but this is not an issue for the majority of cases.

Normality Test Output

ARCH Effect Test

This test often arises in examining or modeling the heteroskedasticity (i.e. fat-tails) in a time series. This is very similar to the white-noise test, the exception being the use of the quadratic (squared) values of the time series. In effect, it examines the non-linear dependency between the lagged values.

$x_t=r_t^2$

$H_{o}: \rho_{1}=\rho_{2}=...=\rho_{p-1}=\rho_{p}=0$

$H_{1}: \exists\rho_{k}\neq 0; k \in [1, p]$

Where

$H_o$ : Null hypothesis (No ARCH Effect)

$H_1$ : Alternative hypothesis (ARCH Effect)

$\rho_k$ : Autocorrelation function for k lags

$p$ : Upper lag limit of the test

ARCH Effect Test Output

Summary

Using NumXL functions, we described the statistical properties of the underlying distribution of our time series data set. We also performed a series of statistical tests on the normality distribution, linear, and non-linear time interdependencies of the data set.

Although these tests may initially sound like an academic curiosity, they are critical to our modeling, especially as we examine the fitted model assumptions and perform residuals diagnosis. We will visit those tests later on in this tutorial.