L6 - Interpreting the OLS regression

Question 1

What happens to the t-distribution for the OLS estimator when the sample become larger?

Answer 1

As the sample size becomes larger the t-distribution becomes more like the normal distribution.

Question 2

What happens to the variance for the OLS estimator when the sample become larger?

Answer 2

As the sample size become very large the variance of the OLS estimator will tend to zero.

V(β(hat)) = (σ_u²)/(Σ^Ni=1(X_i-X(bar))²) –> 0 as N –> ∞

as N gets larger the positive dominator (due to its squared nature) gets larger thus tending towards 0

Question 3

What happens when the sample of the OLS estimator gets too large?

Answer 3

The asymptotic distribution of the OLS estimator is degenerate i.e. the PDF collapses onto a single value as the sample size becomes arbitrarily large

Question 4

How does increasing the sample size affect the OLS estimator PDF on a graph?

Answer 4

• gets thinner and tall around the mean get closer and closer to a vertical line.

Question 5

How can we compare estimators on the basis of their asymptotic variance?

Answer 5

Sometimes we wish to compare estimators on the basis of their-asymptotic variance (why?) - to see if the estimator is consistent –> this being that is the number of observations tended to infinity would we get the population estimator, or the difference between the estimate and the actual value is so small that is is basically zero

but we can’t do this if their distributions are degenerate.

• Instead we look at a transformation of the distribution which produces a non-zero finite variance.

sqrt(N)(β(hat)-β) ~N(0, (σ_u²)/(1/N *Σ^N_i=1(X_i-X(bar))²)

As N –> ∞

sqrt(N)(β(hat)-β) –> ~N(0, (σ_u²)/(σ_X²))

Question 6

What is the OLS Regression Residuals?

Answer 6

The OLS regression residuals can be defined as the difference between the actual values of Y and the fitted values from the regression equation.

u_i(hat)= Y_i-α(hat)-β(hat)X_i

• The regression residuals are not the same thing as the equation errors which are unobservable –> as in a general regression formual we don’t know the value of α or β
• The residual sum of squares from the regression modelcan be written as:

RSS= Σ^N_i=1(u_i(hat))²

• Where as the equation errors are the vertical distances from the scatter plot to the population regression function.
• The regression residuals are the vertical distances from the scatter plot to the sample regression function.

Question 7

What is the 1st important property of the OLS residuals?

Answer 7

the OLS residuals sum to zero:

Σu_i(hat)= Σ(Y_i-α(hat)-β(hat)X_i)

= ΣY_i-nα(hat)-β(hat)ΣX_i

= N(Y(bar)-α(hat)-β(hat)X(bar)) = 0

by virtue of the property that the OLS regression passes through the sample means of the data. –> Y(bar)-α(hat)-β(hat)X(bar) is what we used to calculate the RSS which is equal to 0

• This is not true of the equation errors. Suppose u~N(0,σ_u²), it follows that:

W= Σ^N_i=1(u_i)~ N(0,Nσ_u²)

• as Nσ_u² there will be some point that are different from 0

Question 8

What is the 2nd important property of the OLS residuals?

Answer 8

The OLS residuals are uncorrelated with the X variable. Note:

u_i(hat)= (Y_i-α(hat)-β(hat)X_i)

sub in for α(hat)

= Y_i-Y(bar)+β(hat)X(bar)- β(hat)X_i

₌ Y_i-Y(bar)-β(hat)(X_i-X(bar))

Therefore:

Σ(X_i-X(bar))u_i(hat) = Σ(X_i-X(bar))(Y_i-Y(bar) -β(hat)Σ(X_i-X(bar))²

Recall that β(hat) =Σ(X_i-X(bar))(Y_i-Y(bar)/Σ(X_i-X(bar))²

Therefore:

cov(X,u(hat))= (1/N-1)Σ(X_i-X(bar))u_i(hat)= 0

Note cov(X,u) = 0 may or may not be true. This is a matter of assumption rather than a mathematical property of the model.

Question 9

What is Covariance?

Answer 9

• Covariance is a measure of how changes in one variable are associated with changes in a second variable.
• Specifically, covariance measures the degree to which two variables are linearly associated. However, it is also often used informally as a general measure of how monotonically related two variables are

Question 10

How can we interpret the Slope Coefficient of the OLS regression model?

Answer 10

• The slope coefficient gives the marginal effect on the endogenous variable of an increase in the exogenous variable
• The interpretation of the slope coefficient depends on the units of measurement.
• in most cases we use constant or real prices as we must take inflation into account to produce the best estimator

Question 11

Why do we use Log-linear equations for an OLS regression?

Answer 11

The slope coefficient of a log-linear regression gives us the elasticity of y with respect to x.

• so if there was a 1% percent increase in x it will give us the incease/decrease in y as a percentage

Note:

dLn(Y_t)/dY_t=1/Y_t

dLn(X_t)/dX_t=1/X_t

dLn(Y_t)/dLn(X_t)= (dY_t/dX_t) * (Y_t/X_t) = η_Y,X

Question 12

How do we interpret the intercept of the OLS regression?

Answer 12

• The intercept is chosen so that the regression line passes through the sample means of the data
• It does not make much sense to think of the intercept as the value of Y when X is zero.
• Although this is mathematically true, the zero value of X often lies well outside the range of the data –> cant be explained by econometrics

Question 13

How do you Predict Error Variance?

Answer 13

• Suppose we wish to predict Y for a given value of X.

Y_p = βX_p+ u_p

Y_p(hat) = β(hat)X_p

Y_p - Y_p(hat) = (β -β(hat)) X_p+ u_p )

If we take the expect value of both sides of the equation, as E(β(hat)) = β and E(Y_p - Y_p(hat))= 0 –> since the OLS estimates of the intercept and the slope are unbiased and the expected value of the error is zero.

Therefore, the prediction error for Y_p is partly due to the random disturbance u and partly due to the inaccurate estimate of β.

V(Y_p - Y_p(hat)) = (β -β(hat)) X_p+ u_p)² = ((σ_u²X_p²)/(ΣX_i²)) + σ_u²

The prediction error variance is smallest when X equals its mean value (in this case zero). It gets larger the further X is from its mean.

• What this means is that we are more likely to get accurate predictions when the value of the x variable used is a ‘typical’ value in the sense that it is close to the sample mean.
• The more extreme the value of x we use, i.e. the further from the sample mean, then the less reliable will be the prediction. The importance of this effect will vary depending on the nature of the data used

Question 14

Why is differencing useful?

Answer 14

• Economic variables often contain a trend. Differencing will remove the trend from the data.
• Regressions in levels may give an unrealistic impression of the explanatory power of equations when the data is trended.
• you can still see if there is a relationship between related variables
• Differencing or taking the difference of the logarithms helps because we are effectively regressing the growth rate on one variable on another
• An equation estimated in the first differences of the logarithms of the variables constitutes a relationship in growth rates:
• dLn(y_t)/dt = (1/y_t) *(dy_t/dt)
• g_I,t= α + βg_Y,t + u_t(hat)

Question 15

What is a summary of all the regressions so far?

Answer 15

1: Y_i= α(hat)+β(hat)X_i + u_i(hat)

Marginal effect of Y on X

2: Ln(Y_i)= α(hat)+β(hat)Ln(X_i) + u_i(hat)

elasticity effect of Y on X –> direct relationship

3. d(Y_i)= α(hat)+β(hat)d(X_i) + u_i(hat)

marginal effect of Y on X

4. dLn(Y_i)= α(hat)+β(hat)dLn(X_i) + u_i(hat)

growth rate effect of Y on X

Question 16

When can you not take logs of a regression?

Answer 16

If even on of the values is negative as Ln is not defined at negative values