5.) Hypothesis Testing Flashcards
The F-Test is …if th
a formal hypothesis test that is designed to deal with a null hypothesis that contains multiple hypotheses or a single hypothesis about a group of coefficients
The first step in the F-test is…
translate the particular null hypothesis in question into constraints that will be placed on the equation
A constrained equation can be thought of as…
what the equation would look like if the null hypothesis were correct
For a constrained equation you…
substitute the hypothesized values into the regression equation in order to see what would happen if the equation were constrained to agree with the null hypotest
In the F-test the null hypothesis always leads to a…
constrained equations, even if this violates our standard practice that the alternative hypothesis contains what we expect is true
The second step in an F-test is to…
estimate this constrained equation with OLS and compare the fit of this constrained equation with the fit of the unconstrained equation
If the fits of the constrained equation and the unconstrained equation are not significantly differerent…
the null hypothesis should not be rejected
RSS =
Residual sum of squares from the unconstrained equation
RSS(M) =
residual sum of squares form the constrained equation
M =
number of constraints place don the equation (usually equal to the number of betas eliminated from the unconstrained equation)
(N-K-1) =
degrees of freedom in the unconstrained equation
F =
((RSSm-RSS)/M) / (RSS/(N-K-1))
RSS(M) is always …
greater than or equal to RSS
Imposing constraints on the coefficients instead of allowing OLS to select their values …
can never decrease the summed squared residuals
As the difference between the constrained coefficients and the unconstrained coefficients increases…
the data indicate that the null hypothesis is less likely to be true
The decision rule to use in the F-test is …
to reject null hypothesis if the calculated F-value from Equation 11 is greater than the appropriate critical F-value
The F-statistic has two types of degrees of freedom…
- ) The degrees of freedom for the numerator of Equation 11 (M, the number of constraints implied by the null hypothesis)
- ) The degrees of freedom for the denominator of the F equation (N-K-1 , the degrees of freedom in the regression equation)
Underlying Principle here is that…
if the calculated F-value (or F-ratio) is greater than the critical value
F-test of overall significance is really testing…
the null hypothesis that the fit of the equation isn’t significantly better than that provided by using the mean alone.
The null hypothesis in an F-test of overall significance
is that all the slope coefficients in the equation equal zero simultaneously
Our decision rule tells us to…
reject the null hypothesis if the calculated F-value is greater than the critical F-value
Seasonal Dummies are…
dummy variables that are used to account for seasonal variation in time-series models
To test the hypothesis of significant seasonality in the data…
one must test the hypothesis that all the dummies equal zero simultaneously rather than test the dummies one at a time.
What are the two kids of errors we can make in hypothesis testing?
Type I: We reject a true null hypothesis.
Type II: We do not reject a false null hypothesis
A decision rule is…
a method of deciding whether to rejct a null hypothesis
A decision rule involves comparing…
a sample statistic with a pre-selected critical value found in talbes
A critical value is…
a value that divides the “acceptance” region from the rejection region when testing a null hypothesis
Decreasing the chance of a Type I Error means…
increasing the chance of a Type II Error
A critical t-value is..
the value that distinguishes the “acceptance” region from the rejection region.
The level of Type I Error in a hypothesis test is also called…
the level of significance of that test
The level of significance indicates the probability of observing an…
estimated t-value greater than the critical t-value if the null hypothesis were correct. It measures the amount of Type I Error implied by a particular critical t-value.
An extremely low level of significance also dramatically increases…
the probability of making a Type II Error
A confidence interval is…
a range that contains the true value of an item a specified percentage of the time. This percentage is the level of confidence associated with the level of significance used to choose the critical t-value in the interval.
A p-value for a t-score is…
the probability of observing a t-score that size or larger (in absolute value) if the null hypothesis were true.
Graphically, the p-value is…
the area under the curve of the t-distribution between the actual t-score and infinity
A p-value is a probability so…
it runs form 0-1. It tells us the lowest level of significance at which we could reject the null hypothesis
The p-value decision rule is…
reject the null hypothesis if the p-value
The most common use of a one-sided t-test is to determine whether a regression coefficient is…
significantly different from zero in the direction predicted by theory
Using a one-sided t-test to in order to be able to control the amount of Type I Error me make, the implication is
Alternative Hypothesis: B > 0
Null Hypothesis: B less than or equal 0
(If positive coefficient predicted by theory)
The Four Steps to use when working with the t-test are..
- ) Set up the Null and Alternative Hypothesis
- ) Choose a level of significance and therefore a critical t-value
- ) Run the regression and obtain an estimated t-value (or t-score)
- ) Apply the decision rule by comparing the calculated t-value with the critical t-value in order to reject or not reject the null hypotheses
The kinds of circumstances that call for a two-sided test fall into two categories…
- ) Two sided tests of whether an estimated coefficient is significantly different from zero.
- ) Two-sided tests of whether an estimated coefficient is significantly different from a specific nonzero value
The t-Test does not test…
- ) Theoretical Validity
- ) “Importance” of an independent variable
- ) I not intended for tests of the entire population
the t-test tests hypotheses about…
individual coefficients from regression equations.
