Hypothesis Testing Steps in Hypothesis Testing: 1. State the hypotheses 2. Identify the test statistic and its probability distribution 3. Specify the significance level 4. State the decision rule 5. Collect the data and perform the calculations 6. Make the statistical decision 7. Make the economic or investment decision Two-Tailed Test (Z-test @ 5%) Null hypothesis:  = 0 Alternative hypothesis:   0 where 0 is the hypothesised mean 1.96 SE 0 Rejection area Rejection area One-Tailed Test (Z-test @ 5%) Rejection area Null hypothesis:   0 Alternative hypothesis:  > 0 1.645 SE 0

Hypothesis Testing – Test Statistic & Errors Test Concerning a Single Mean Type I and Type II Errors Type I error is rejecting the null when it is true. Probability = significance level. Type II error is failing to reject the null when it is false. The power of a test is the probability of correctly rejecting the null (i.e. rejecting the null when it is false) Do not reject null Reject null Correct Type I error Type II error Decision H0 true H0 false

Hypothesis about Two Population Means Normally distributed populations and independent samples Examples of hypotheses: Population variances unknown but assumed to be equal s2 is a pooled estimator of the common variance Degrees of freedom = (n1 + n2 - 2) Population variances unknown and cannot be assumed equal

Hypothesis about Two Population Means Normally distributed populations and samples that are not independent - “Paired comparisons test” Possible hypotheses: Symbols and other formula Application The data is arranged in paired observations Paired observations are observations that are dependent because they have something in common E.g. dividend payout of companies before and after a change in tax law

Hypothesis about a Single Population Variance Possible hypotheses: Assuming normal population Symbols s2 = variance of the sample data 02 = hypothesized value of the population variance n = sample size Degrees of freedom = n – 1 NB: For one-tailed test use  or (1 – ) depending on whether it is a right-tail or left-tail test. Chi-square distribution is asymmetrical and bounded below by 0 Obtained from the Chi-square tables. (df, 1 - /2 ) Obtained from the Chi-square tables. (df, /2) Lower critical value Higher critical value Reject H0 Fail to reject H0 Reject H0

Hypothesis about Variances of Two Populations Possible hypotheses: Assuming normal populations The convention is to always put the larger variance on top Degrees of freedom: numerator = n1 - 1, denominator = n2 - 1 F Distributions are asymmetrical and bounded below by 0 Obtained from the F-distribution table for:  - one tailed test /2 - two tailed test Critical value Fail to reject H0 Reject H0

Correlation Analysis Sample Covariance and Correlation Coefficient Correlation coefficient measures the direction and extent of linear association between two variables Scatter Plots x y x Testing the Significance of the Correlation Coefficient Set Ho:  = 0, and Ha:  ≠ 0 Reject null if |test statistic| > critical t Degrees of freedom = (n - 2)

Parametric and nonparametric tests rely on assumptions regarding the distribution of the population, and are specific to population parameters. All tests covered on the previous slides are examples of parametric tests. Nonparametric tests: either do not consider a particular population parameter, or make few assumptions about the population that is sampled. Used primarily in three situations: when the data do not meet distributional assumptions when the data are given in ranks when the hypothesis being addressed does not concern a parameter (e.g. is a sample random or not?)

Linear Regression Yi Xi Basic idea: a linear relationship between two variables, X and Y. Note that the standard error of estimate (SEE) is in the same units as ‘Y’ and hence should be viewed relative to ‘Y’. x Y, dependent variable Xi i error term or residual X, independent variable Yi Mean of i values = 0 Least squares regression finds the straight line that minimises

The Components of Total Variation

ANOVA, Standard Error of Estimate & R2 Sum of squares regression (SSR) Sum of squared errors (SSE) Sum of squares total (SST) Standard Error of Estimate Coefficient of determination R2 is the proportion of the total variation in y that is explained by the variation in x Interpretation When correlation is strong (weak, i.e. near to zero) R2 is high (low) Standard error of the estimate is low (high)

Assumptions & Limitations of Regression Analysis The relationship between the dependent variable, Y, and the independent variable, X, is linear The independent variable, X, is not random The expected value of the error term is 0 The variance of the error term is the same for all observations (homoskedasticity) The error term is uncorrelated across observations (i.e. no autocorrelation) The error term is normally distributed Limitations Regression relations change over time (non-stationarity) If assumptions are not valid, the interpretation and tests of hypothesis are not valid When any of the assumptions underlying linear regression are violated, we cannot rely on the parameter estimates, test statistics, or point and interval forecasts from the regression