# Chapter 9 Hypothesis Testing in the GWN Model

Updated: May 11, 2021

The core of statistical analysis involves estimation and inference. The previous chapters considered estimation of the GWN model parameters, and the computation of standard error estimates and confidence intervals. In this chapter we discuss issues of statistical inference within the GWN model. Statistical inference is about asking questions about a model and using statistical procedures to obtain answers to these questions with a given level of confidence. For example, we might find that our estimate of the monthly mean of an asset return is positive but due to large estimation error we might ask the question: is the true monthly mean return positive? As another example, we might find that a normal qq-plot of returns shows some evidence of non-normality and we ask the question: are returns normally distributed? As a third example, we might find that rolling estimates of the mean and volatility show some evidence of non-constant behavior and we might ask the question: are returns covariance stationary? Statistical hypothesis testing gives us a rigorous framework for answering such questions with a given level of confidence.

The R packages used in this chapter are IntroCompFinR, PerformanceAnalytics, tseries, and zoo. Make sure this packages are installed and loaded before replicating the examples.

1. Hypothesis Testing in the GWN Model
1. Specification tests
1. Normal distribution
1. Tests for skewness and kurtosis
2. JB test
3. What to do if returns are not normally distributed?
2. No autocorrelation A. individual and joint tests
3. Constant parameters (covariance stationarity) A. Formal tests: testing same mean, volatility in sub-samples B. Informal diagnostics - standard error bands on rolling estimators
2. Using Monte Carlo Simulation to Understand Hypothesis tests
1. Size
2. Power
3. Using the bootstrap for hypothesis tests
1. exploit duality between hypothesis tests and confidence intervals
2. Appendix: Distributions for test statistics
1. Student’s t distribution
2. Chi-square distribution