3. Testing Hypotheses about Single Coefficients Flashcards
Null hypotheses
The original hypothesis which we are challenging
Alternative hypothesis
The hypothesis which we are instead putting forward
What do we need to draw perfect statistical inferences?
We need to know everything about the sampling distribution of our estimators
When do we know everything about the sampling distribution of our estimators?
If we know their means and variances and the distribution is normal
What is MLR6?
The population disturbances (u) are independent of the explanatory variables and are normally distributed with zero means and variance ó^2
When do we reject Ho?
When the probability of getting the estimate is less than the significance level
Why do we use the t distribution?
Because it takes into account the fact we have estimated ó^2 and it takes into account how much info we used
How many degrees of freedom do we get?
n-k-1
When can we draw inferences about causal relationships from OLS estimates?
When the CLM assumptions are valid
P value
The exact significance level at which an estimate ceases to be significantly significant
In a two sided test with 5% sig level, if the p value is 3% is there sufficient evidence to reject Ho?
Yes because the p value is split each end so it will actually be in the 1.5th percentile at each end
Type 1 error
When we reject a true null, the probability of this happening is equal to the significance level
Type 2 error
When we fail to reject a false null
What is the power of a test
Power=1- prob(type 2 error)
How are type 1 and type 2 errors related?
They are inversely related
Confidence intervals
A range of values where a certain % of values will fall. E.g for a 5% significance level 95% of values will fall here
SSE
Explained sum of squares. Measures the variation in ŷi, I.e the variation in yi that is explained by the model
SSR
Residual sum of squares. Measures the variation in ûi, I.e. the variation in yi that is unexplained
Equation for R^2
R^2= SSE/SST= 1-SSR/SST
What is R^2?
It is the ratio of the explained variation to the total variation. Written in other words it is the fraction of the sample variation in y that is explained by all of x
Evaluate the effectiveness of R^2
- a higher R^2 shows a better fit of the model to the data
- can be used to compare models with the same dependent variable
- adding an extra variable will increase R^2 even if it doesn’t improve the model
Adjusted R^2 formula
Adjusted R^2= 1-((1-R^2) x (n-1)/(n-k-1)
Evaluate the adjusted R^2
- adjusted R^2< R^2
- as k increases, the size of the adjustment increases
- it is used to compare models with different k
- the adjustment is arbitrary and the value can rise when a statistically insignificant regressor is added to the model
