OLS

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.

It suggests that an asset with a higher perceived risk would pay a higher return on average. In other words the mean return would be related to the variance of the return.

Negative surprises seem ti increase volatility more than positive surprises.

VaR measures the probability that a portfolio will face ist worse outcome.

The models can be used to estimate the dynamic relationship amongst variables; These models are used heavily for short-term forecasting.

Each variable depends on the lags of all variables included in the VAR; It is stationary if ist mean and variance are constant over time.

It is chosen such that the value of AIC and SIC are minimised; AIC tends to favour model with longer lags and SIC returns the mosta parsimonious model as it penalises extra estimated parameters more heavily.

If the lagged values of one variable have no explanatory power for all variables in the VAR, then this variable is said to be weakly exogenous; Granger causality can occur because a variable left out of the VAR is correlated with the variables included. So results should be interpreted with caution.

Can be used to measure how relationships between variables changes over time; Used to estimate stochastic volatility models that are central to many financial theories; Decompose time-series into a cycle and trend component; used to determine if a common component is driving a group of time-series; Estimate missing observations.

Provides estimates of the state given current information; one may be interested in estimating the state vectir at date t based on information contained in the entire dataset.

Note that beta stands for the change in H when Y changes by 1 unit. Note that this is not the elasticity.

Distance between line and data points is minimised.

Regression with multiple variables

y = xb + e

Independent variable

The coefficient

The residuals or error term

Linearity, Full rank, Regressors are exogenous, Homoskedasticity and no autocorrelation, Non-stochastic regressors, Normal disturbances

The model species a linear relationship between y and x

x has full column rank. There is no exact linear relationship between the di¤erent independent variables

E (e\x) = 0 The mean of the residuals (holding x xed) is zero.

The residuals have a fixed variance sigma^2 and the residual for observation i is uncorrelated with the residual for observation j.

x is non-stochastic as in experimental data.The conometricianchooses the values of the regressors before observing y. For example, where x is fertilizer and irrigation and y is agricultural yield. In this case, we do not need to condition on x in the assumptions discussed above

We assume that the residual is normally distributed e\x ~ N(0,sigma^2*I)