Econometrics

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[edit] Econometrics 1

This is a graduate level course concerned with theory and application of linear regression methods. The classical regression model is discussed, and the statistical properties of the estimator are examined. The effect of violations of the classical assumptions are considered, and appropriate estimation methods are introduced. This course is the first of a two-course sequence. At course completion, a successful student will:

  • understand the statistical foundations of the classical regression model.
  • be able to explain the properties of the least-squares estimator and related test statistics.
  • be able to apply these methods to data and interpret the results.

[edit] Recommended texts

-Davidson and MacKinnon "Econometric Theory and Methods"

-DeGroot and Schervish "Probability and Statistics" 3rd edition

A review of matrix algebra is recommended. In econometrics, it is necessary to work with very large sets of data. In order to manipulate the data and follow the discussion, you must be familiar with matrices.

[edit] Random variables

How do we handle random processes? We can define a random variable as a measurable function defined on a probability space. Bievens(2004)

See set theory. Kolmorgorov gives us the axioms of probability.

  1. Firstly, the probability of some outcome, A, is greater than or equal to zero, P(A)\ge0 for A contained in the probability space S A\in S.
  2. Secondly, the probability over the sample space is equal to one, P(S) = 1 \,.
  1. Thirdly, the probability of A and B is equal to the probability of A plus the probability of B, if A and B are disjoint and contained in the sample space S P(A+B)=P(A)+P(B); A\cap B=\oslash; A,B \in S.

For example, in a coin toss, the probability space is heads and tails. If x is a function over the probability space, we can say that x(heads) equals one and x(tails) equals zero. So the probability that x is one equals the probability of heads, and the probability that x is zero equals the probability of tails, and they are disjoint probabilities. So if the probability of heads is Ph, the probability of tails is 1-Ph. (Note that this example does not assume that you have a fair coin, though you could.)

Matrices: positive semidefinite (PSD), positive definite (PD), quadratic forms, symettric, idempotent, diagonal, block diagonal,

Matix derivatives

The Probability Distribution Function is F_x(x_0) =P_x(\alpha)\, where \alpha = {x|x\le x_0}



Description of linear regression:

By having a set of y's we are assuming are dependent on a set of x's, we solve for some constant \beta \,. (y is a kx1 vector and x is a kxn matrix multiplied by an nx1 vector \beta\, which relates y and x, where k is the number of observations, and n is the number of predictor or independent variables.)

y = x\beta + \epsilon\,

To solve for \beta\,, which will give us an expected value of \beta\, we will call beta hat \hat\beta\, we use this linear operation:

\hat\beta=(x'x)^{-1}x'y\,

This gives us beta hat, \hat\beta\,.

Then x\hat\beta = \hat y or x times beta hat equals y hat, where y hat is our expected value for y given x and our calculated value of beta, beta hat.

Now the difference between the predicted values of y and the actual values of y are given by: y-x\hat\beta=y-\hat y = e where e\, is residual values, sometimes called error.


Matrix algebra,

Properties of Ordinary Least Squares (OLS estimator beta hat is unbiased for beta, and the Covariance Matrix for beta hat is \sigma ^2 (x'x)^{-1}\,), Classical Normal, The Information Matrix, Chi Square Distribution, The relationship between e and epsilon, The Maximum Likelihood Estimator of the variance is biased, Distribution of the variance under normality,


regression, principles of estimation and testing, stationary time series models, limited dependent variable models, longitudinal (panel) data models, generalized methods of moments, instrumental variable models, non-stationarity, stochastic trends, co-integration,

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