7.2. Regression Analysis II Flashcards
Testing hypotheses, between the two coefficients…
H0: beta (the slope) equals zero and therefore there is no influence of X on Y.
H1: beta (the slope) doesn’t equal zero and therefore there is influence of X on Y.
Compute the test statistic.
Find the critical value (using the two-tailed t-test statistic).
If test statistic > critical value, we reject the null hypothesis and acknowledge a statistically significant difference.
Testing the significance of R2…
Another check of the quality of the regression equation, testing to see if R2 is significantly greater than zero.
This uses F-distribution.
H0: R2 equals zero, implying that beta (the slope) equals zero.
H1: R2 doesn’t equal zero, implying that beta (the slope) doesn’t equal zero.
Compute the test statistic.
Find the critical value, using the F-distribution table.
Prediction…
We can calculate prediction or confidence intervals for data:
- Prediction interval for an individual observation on Y when X = a particular value.
- Confidence interval for the position of the regression line at X = a particular value.
Both confidence and prediction interval estimates are most precise, when…
- Xp=X.
- The closer the sample observations lie to the regression line (i.e. the smaller the standard error).
- The greater the spread of sample X values (i.e. larger (Xi-X)2).
- The larger the sample size.
For Regression Analysis Part II (single regression)…
Please see 7.3. Casino Expansion Example.
Multiple regressions…
We have more than one explanatory (independent) variable. This is multivariate regression.
Single regression is very limited.
Generally:
- The principle is to fit a plane, not a line as in the simple regression.
- This minimises the sum of squares of vertical distances from each point to the plane.
A regression model with two explanatory variables…
b0: signifies the constant, or intercept, on the Y axis.
b1: is the slope of the plane in the direction of X1 axis.
b1: shows the effect of a unit change in X1 on Y, assuming X2 remains constant.
b2: is the slope of the plane in the direction of X2 axis.
b2: shows the effect of a unit change in X2 on Y, assuming X1 remains constant.
If both X1 and X2 change by 1, the effect on Y is b1+b2.
Data transformation…
We need to transform nominal data to adjust it for inflation.
To calculate real income, we must divide GDP by a GDP deflator.
To calculate real import prices, we must divide the price of imports by the retail price index.
Interpreting results…
n(gdp) can be calculated which shows the elasticity of one variable (imports) with respect to the other variable (GDP).
Imagine the value is 2.14. This means that a 1% rise in the latter variable (GDP) leads to a 2.14% rise in the former variable (imports).
Using elasticity, we can use percentages rather than simply ‘units’ making our interpretations more accurate.
By a similar calculation, we can calculate this for the other way:
For example, a 1% rise in import prices leads to a 0.03% rise in import demand.
Improving the model, using logarithms…
We might need to find more explanatory variables.
We might need to try lagged variables as explanatory variables.
We might need to try a different functional form for the equations.
The use of natural logarithms:
- Non-linear transformation of the data.
- More direct estimates of the elasticities.
- Common practice.
Issues with multivariate regression…
Autocorrelation: when one error observation is correlated with an earlier one.
Multicollinearity: when some or all of the explanatory variables are highly correlated.
Omitted variable bias: bias introduced in the estimated coefficients of independent variables due to the omission of relevant variables from the regression model.
