1. 1 MADE Why do we need econometrics? If there are two points and we want to know what relation describes that? X Y

2. 2 MADE Why do we need econometrics? But if there’s more than just two points for two variables?

3. 3 MADE Why do we need econometrics? How would we look for this line? MINIMISING THE RESIDUALS!!!!

4. 4 MADE What is an econometric model?  Some things about reality are known… –GDP per capita –capital accumulation –volume of trade  … but the relations between them are unknown –correlation –causality  we need a tool to seek the latter using the former  Costs? We need to simplify the reality

5. 5 MADE An example of a model Suppose you wanted to see what is the degree of gender discrimination in wages. Your model: wages=f (gender and ???) –education –experience –profession –city/rural area –… We cannot consider everything because: –no data –model quality => STATISTICS

6. 6 MADE Random versus deterministic What is a variable? What is a random variable? –example: height of all the people in this room Can you ever get a deterministic number from a random one? What is EXPECTED VALUE? –for a deterministic variable –for a random variable

7. 7 MADE Are residuals form this graph random or deterministic?

8. 8 MADE An example of a model revisited Let’s go back to the example of gender discrimination: We said the model was like this wages =f (gender and ???) But now we know that in fact: wages = constant + coeff*education + coeff*experience + coeff*gender + coeff*whatevereslewethinkof + residuals We don’t know the coefficients => we seek a method to find them!!! Residuals depend on how we choose the coefficients and are unknown (random)

9. 9 MADE Finding a method We want to minimise our „error”: or

10. 10 MADE Finding a method We can write each of the elements as :

11. 11 MADE Finding a method What we have is: –X – a matrix of exogenous (input) variables („knowns”) –y - a vector of the endogenous (but still input) variable (we think we know the results of the random process) –ɛ – unknown residuals that can be only estimated using residuals from the model –β – unknown parameters that we want to estimate (output) What we need is: –a model that will let us know β’s, with ɛ’s as small as only possible

12. 12 MADE Finding a method Let’s define: Where: is a theoretical, fitted value of y’s »e’s are only estimates of ɛ’s, but do not have to be equal »b’s are only estimats of β’s, but are chosen such that, y and y hat are as close as possible

13. 13 MADE Finding a method We find the method for estimation by minimising the residuals, but: –There is a lot of them –They can be very big (positive and negative) and still add up to zero => we need to take squares (distances) and not direct values

14. 14 MADE Finding a method We look for the first order conditions for: So we differentiate and put equal to zero:

15. 15 MADE Finding a method When it comes to matrices, multiplication is no longer as straightforward (it matters what comes first and you can’t divide) What you can is pre-multiply by an inverted matrix In order for a matrix to be invertible, it has to be nonsingular (no row and no column is a linear combination of the others) X’X is a matrix seems to meet these conditions

16. 16 MADE Finding a method We have an optimum, but we don’t know if it’s a max or a min => need to find second derivative and prove it’s positive to be sure to have a minimum (so residuals as small as possible) It is positive, so we have found what we were looking for

17. 17 MADE Properties of OLS 1.X’e=0 2.Fitted and actual values of y are on average equal 3.Σe=0 (for a model with a constant) 4.There is nothing more systematic about y than already explained by X (fitted y and residuals are not correlated)

18. 18 MADE Properties of OLS If a model has a constant… … and then

19. 19 MADE Is OLS the best? Can we be sure that OLS will always give us the best possible estimator? If assumptions are fulfilled, OLS is BLUE (meaning Best Linear Unbiased Estimator) Assumptions: 1.y=X β 2.X is deterministic and exogenous 3.E(ɛ i )=0 4.Cov(ɛ i,ɛ j )=0 5.Var(ɛ i )=σ 2 What do we loose on linear and unbiased?

20. 20 MADE Variance-covariance matrix

21. 21 MADE What do we know about OLS properties It is unbiased:

22. 22 MADE What do we know about OLS properties? The variance of the parameters is given by: so we only need to find an estimator of σ, but: so…

23. 23 MADE What do we know about OLS properties? …

24. 24 MADE Why do we need the properties? How can we say that a model is good? –We only know that among linear and unbiased we have estimators of β that yield lowest errors) How can we say if one model is better than other? –So far we didn’t ask this question at all! How can we say AT ALL if a variable really is correlated with another? –So far we only considered setting up a model, but in reality this is an implicit hypothesis and needs to be tested!

25. 25 MADE How good our model is? We can ask how big are the residuals when compared to the input values TSS=ESS+RSS with a constant

26. 26 MADE How good our estimates are? We can test the values we have obtained vis-a-vis a hypothesis that they are zero

27. 27 MADE Preview of coming attractions Hypothesis testing Understanding the output of any statistical package (or tables in papers you have to read ) Interpretation Prognosis