Linear Regression Part 2

What does the log-log form mean and how would you resolve this to create a linear relationship?

- Both dependent and independent variables are logarithmic (relative change of X and relative change of Y) - Take the natural log of both variables to create a linear relationship

What does the lin-log form mean and how would you resolve this to create a linear relationship?

- Dependent is linear - Independent is logarithmic (Meaning there is an absolute change in the dependent variable for a relative change in the independent) Take the natural log of the independent variable to resolve this

What does the log-lin form mean and how would you resolve this to create a linear relationship?

- Dependent variable is logarithmic (hence lnY) - Independent variable is linear To solve this take the natural log of Y (dependent) Slope coefficient in log-lin models is the relative change in the dependent variable for an absolute change in the independent variable.

Data requirements for the F-test?

- Sum of squared error (SSE)(Unexplained variation) - Regression sum of squares (RSS)(Explained variation)

What is the F-test? How is it useful?

- The F-test determines how effective a group of independent variables explain the variation of the dependent variable. The larger the F-test value the better the independent variable explains the variation of the dependent

Total sum of squares is the blue regression line

- The difference between the mean and the predicted Y values are shown in the explained sum of squares (Regression sum of squares) - Whilst the unexplained sum of squares is the observations that lie above the line, ie the difference between the actual and mean values for Y.

What is the (RSS) Regression Sum of Squares?

- The explained variation in the dependent variable (aka explained sum of squares) - Calculated by the difference between Y predicted - Y mean squared, and add those numbers together for each value.

What is the Standard Error of Estimate (SEE)?

- The std deviation of the regression residual, basically measuring the fit of the regression line. - Smaller the SEE the better the fit - Although doesn't give suitability of the independent variable in predicting the dependent

What is the Total Sum of Squares (SST)?

- The total variation of the dependent variable - Calculated by the sum of squared differences between the actual y-value and the mean y-observation This can be broken down into Explained variation (RSS) and unexplained variation (SSE)

What is the (SSE) Sum of squares errors? (Also known as residual sum of squares)

- The variation of the dependent variable unexplained by the independent variable. - Calculated by squaring the differences between Yactual - Y predicted and summing all those together.

How do you select the correct functional form?

1) Calculate the coefficient of determination (R^2) a higher r squared the better 2) F-statistic, higher values are better 3) Standard error of the estimate, a lower standard error is better

What are the three functional forms of simple linear regression?

1) Log-Lin 2) Lin-Log 3) These always follow the form of dependent variable first then independent next, so 1) means the dependent variable is logarithmic and the independent is linear

What is analysis of variance (ANOVA)?

A statistical procedure used to split the total variability of a variable into components that can be ascribed to different sources - Determines the effectiveness of the independent variable in explaining the variation in the dependent variable

How would you calculate the predicted dependent variable given an estimated linear regression model and a value for the independent variable

Basically just insert the value of X (independent variable) slope (b1) and Y intercept (bo) to get the Y value

Null hypothesis: Slope coefficient is not different from zero Alternative: Slope coefficient is different from zero T-test stat= (2.05 - 0)/ 0.3= 8.33 We use 0 as B1 because we are testing if the slope coefficient/ Beta is significantly different from zero

Critical value to compare 8.33 against is 2.009 5% significance level, so use 0.05 on the two tail area on top df= n-2= 52-2= 50 Critical value = 2.009 8.33 > 2.009, therefore we can reject the null, slope coefficient is statistically different from zero.

Do you reject or accept the null at a 5% significance level using the table attached

F-test is a one tailed test and our value of 18.89 exceeds 7.71 therefore we reject the null and fail to reject the alternative hypothesis. - Therefore there is some relationship between the dependent and independent variables. - Meaning we can conclude the slope coefficient is significantly different from zero. df on numerator and denominator - In the numerator we have k, number of independent variables, this is 1 for our example, so use this as df1 - The the denominator we have df= n-2= 4, so use this from the left-hand side

Predicted value (Y)= 0.01702 + (0.1927 x 0.20) Y= 0.05556 or 5.56% Std error is given at 0.007615 Critical values = 95% confidence interval and df = 4 so critical value is 2.776 (see flip-side for t distribution table) df = n - 2 n was 6 countries

Interval = 5.56% +/- (0.007615 x 2.776) = 5.56% +/- 0.02113 = 3.47% <-> 7.67% Inflation will be within these values with a 95% confidence when money supply growth is 20%

How would you make an confidence interval of these predicted values

Just take the predicted value and +/- (critical t-value with n-2 df x std error)

Calculate the F-statistic from this table?

Note: N=6

What is the decision point of the F-test? and is the F-test a one or two tailed test?

Reject the null hypothesis if the F-value is greater than the critical value F-test is a one tailed test. (Usually used when you have more than one independent variable, if you have one variable it'd give the same value as a t-statistic)

How would you calculate a confidence interval around a particular slope coefficient?

Same plus and minus format as the predicted Y value Ho (Null)= b1= 0 Ha (Alternative)= b1 not equal to 0 Reject null if confidence interval excludes 0

A question like this in the exam would usually only ask for about 4= n

Take this sum = (0.000106704 / n - 2)^1/2 SEE= (0.000106704 / 6 - 2)^1/2= 0.00516 Remember: ^1/2 also means square root

What is the coefficient of determination?

The coefficient of determination is R^2

Total variation=?

Total variation= Explained Variation (RSS) + Unexplained Variation (SSE)

Basically all you need to do when you have these different forms that are none-linear is to use the natural logarithmic of one or both variables, which will transform the relationship to a linear relationship

Very simple

T-tests can be two tailed, therefore will use a confidence interval, must do the +/-

What is the t-value here?

Y= Inflation X= Money Supply Y= 0.01702 + 0.1927X What is the inflation rate if money supply is at 10%?

Y= 0.01702 + (0.1927 x 10%) Y= 0.03629 Inflation is expected to be 3.629% when inflation is 10%