OLS

Whether the R-hat^2 rises or falls with an additional regressor depends on whether the improvement in t due to the additional regressor more than o¤sets the correction for the loss of an additional degree of freedom.

In a linear regression model, the least squares estimator ˆb is the minimum variance linear unbiased estimator of b. The OLS estimator is the most efficient in the class of linear unbiased estimators. Other unbiased estimators may exist but they have a larger variance. Requires assumptions A1 to A4, but not normality.

^b is a constant estimator of b as the number of observations reaches infinity; If the regressors are well behaved and observations are independant then asymptotic normality of the least squares estimator does not depend on normality of disturbances.

If disturbances are normally distributed the OLS estimator is also the maximum likelihood estimator (MLE). Means that OLS is asymptotically e¢ cient among consistent and normally distributed estimators. Large sample counterpart to Gauss Markov (Cramer Rao Lower bound).

The object of interval estimation is to present an estimate of the parameter with a measure of uncertainty attached to iti.e. b-hat +- sampling variability

Wald Test: The Wald test measures how close Rb - q is to zero

As the sample size grows, the t-distribution approaches the normal distribution and the F-distribution approaches the Chi-squared from above. This suggests that in moderate samples, the t and the f distributions provide a conservative approximation.

The variance of the error term is not constant over time.

Hypothesis testing is the dame in GLS although there is no counterpart to R^2 in GLS because the variables are transformed. We could test it on the original data but the R

Unbiased; Conistent; Inefficient relative to GLS; Standard errors are not reliable.

Those are the standard errors which account for heteroskedasticity. In small samples it may provide standard errir that are biased downwards.

A test for heteroskedasticity. Tests if the individual variance is constant.

The Breusch-Pagan (BP) test is one of the most common tests for heteroskedasticity. It begins by allowing the heteroskedasticity process to be a function of one or more of your independent variables, and it’s usually applied by assuming that heteroskedasticity may be a linear function of all the independent variables in the model.

We assume that Omega is known and then compute the GLS estimator.

FGLS may be inefficient if the form of heteroskedasticity is incorrectly specified; Use OLS with white standard errors; Scale variables appropriately; Models for heteroskedasticity.

Residuals are correlated over time e.g. due to missing variables in the regression that are correlated across time. Assume homoskedasticity; OLS in inefficient; GLS and FGLS required to circumvent this problem.

OLS is biased and inconsistent if there is a lagged dependent variable. This happens because in this model the covariance between our error and the lagged variable is unequal to zero. The exogenity assumption is violated.

A test for autocorrelation. H0: There is no autocorrelation. The test may not be appropriate in the presence of lagged dependant variable.

Tests for autocorrelation. H0: No autocorrelation.

We replace a variable with the lagged variable so it does not suffer from the covariance being unequal to zero and it is highly related to the original variable.

The ramsey reset test. It can be interpreted as a test for linearity.

The RHS variables are highly correlated; Individual coefficients my be indugnificant but jointly significant; Small changes in data may lead to huge changes in coefficient estimates.

White noise; Linked to past values (AR); Moving average (MA); or a combination (ARMA).

Expected value is independent of t; Variance is independent of t; Covariance is a function t-s but not of t and not of s.

Divide both sides by (1-ThetaL) . This is referred to as inverting the MA process to get an AR process. Thisinversion is only possible if the roots of the equation 1-ThetaL = 0 lie outsidethe unit circle. For the MA(1) process the root is 1/Theta. Thus |Theta| < 1 forinveribility.

If the model is AR then OLS can be used; If the model contains MA-term then the model becomes non-linear and OLS cannot be used. ARMA model can be estimated by a maximum likelihood or minimising the squared residuals.; If the error is normal then the two estimators coincide.

Transform yt to obtain stationary series; Estimate an ARIMA(p,q) model; Generate the residuals and check if they are white noise,. If not re-specify the model.

The series is integrades of order 1 (I(1)) if it has to be differenced once to achieve stationarity; Macroeconomic data is generally I(1) they are growing or wandering about with no tendency to return to the mean.

It test if there is a unit root in the data. H0: There is a unit root. Under H0 the t-statistics does not have a t-distribution. The distribution has been tabulated by dickey fuller.

There are laggs added to account for possible autocorrelation in the error term. The number of lags has to be chosen beforehand which can be done using AIC.

There can be more than one cointegrating vector; There can only be M-1 cointegrating relationships between the variable; Number of cointegrating vectors is called the cointegrating rank.

The modely obeys the classical assumptions and OLS is BLUE. However, there exists non-linear estimators that are more efficient.

The GARCH model generalises by adding lags of the variance on the RHS of this equation.

It can be estimated via a maximim likelyhood. It is possible to relax the assumptions that the error is normally distributed. For example, it may be reasonable to assume that the error has T distribution in the case of many financial time series.