Econometrics I: OLS. Dean Fantazzini. Dipartimento di Economia Politica e Metodi Quantitativi. University of Pavia

Transcription

1 Dipartimento di Economia Politica e Metodi Quantitativi University of Pavia

2 Overview of the Lecture 1 st EViews Session I: Convergence in the Solow Model 2

3 Overview of the Lecture 1 st EViews Session I: Convergence in the Solow Model 2 nd EViews Session II: Estimating a money demand equation 2-a

4 Overview of the Lecture 1 st EViews Session I: Convergence in the Solow Model 2 nd EViews Session II: Estimating a money demand equation 3 rd EViews Session III: Monte Carlo Simulation 2-b

5 EViews Session I: Convergence in the Solow Model a) Opening the workfile. EViews is build around the concept of an object. Objects are held in workfiles. Thus open the workfile GROWTH.WMF. Remember that the variables are defined as follows: Variable Definition gdp60 GDP per capita in real terms in 1960 gdp85 GDP per capita in real terms in 1985 pop60 Population in millions in 1960 pop85 Population in millions in 1985 inv Average from 1960 to 1985 of the ratio of real domestic investment to real GDP geetot Average from 1970 to 1985 of the ratio of nominal government expenditure on education to nominal GDP oecd Dummy for OECD member 3

6 EViews Session I: Convergence in the Solow Model... this is the Barro Data Set: Cross-sectional Data Set containing macro information of 118 Countries Collected in 1985 OECD: Organisation for Economic Cooperation and Development 24 members in 1985: Australia, Austria, Belgium, Canada, Denmark, Germany, Finland, France, Great Britain, Greece, Iceland, Ireland, Italy, Japan, Luxembourg, Netherlands, New Zealand, Norway, Portugal, Spain, Sweden, Switzerland, Turkey, USA 4

7 EViews Session I: Convergence in the Solow Model b) Sample The sample is the set of observations to be included in the analysis Often we analyse only subset of this set of observations Example: OECD countries Parameter values are missing for observations smpl

8 EViews Session I: Convergence in the Solow Model c) Generating new series. We need some additional variables for our analysis. Generate the series l gdp60, l gdp85, l pop60 and l pop85 which contain the logarithm of the original series. You can follow two ways : ù Use the GENR button and type l gdp60=log(gdp60), and do the same for the other series. Use the commanding line: series l gdp60 = log(gdp60) Furthermore we need the growth rates of the GDP and the population (GENR - gr gdp=(l gdp85-l gdp60)/25, GENR - gr pop=(l pop85-l pop60)/25). 6

9 EViews Session I: Convergence in the Solow Model Theory (Solow Model): All countries converge against a steady- state for the per capita earnings Poor countries grow faster as rich countries (see also Barro, Sala-i-Matin, Economic growth) d) Analyzing absolute convergence by means of a scatter plot. Plot l gdp60 against gr gdp in a scatter plot. What does the plot tells you about absolute convergence? GR_GDP L_GDP60 Figure 1: Example: Scatter plot 7

10 EViews Session I: Convergence in the Solow Model e) Analyzing absolute convergence by means of a regression. Regress gr gdp on l gdp60 and a constant (click on gr gdp - press STRG and click on l gdp60 - click on one of the series - OPEN EQUATION - OK). Using the online help become acquainted with the regression output (HELP - SEARCH - regression output). What does the regression output tells you about absolute convergence? LS GR GDP c L GDP60 8

11 EViews Session I: Convergence in the Solow Model f) Conditional convergence *. Regress gr gdp on l gdp60, gr pop, inv, geetot and a constant. Interpret the regression output. g) Setting the sample*. One of the important concepts in EViews is the sample of observations. The sample is the set of observations to be included in the analysis. Set the sample to OECD countries. SAMPLE - type in if-box: oecd=1. smpl if OECD=1 9

12 EViews Session I: Convergence in the Solow Model h) Redoing the analysis for OECD countries. Repeat d), e) and f) with the new sample. What changes? Interpret your results. 10

13 EViews Session II: Estimating a money demand equation Some Premises... Detection of Heteroscedasticity: Residual Plot; White Test; Breusch - Pagan test; Detection of Autocorrelation: Residual Plot; Durbin-Watson test (... remember that X have to be deterministic); d=2 no autoc., d<2 positive autoc., d>2 negative autocor. Breusch - Godfrey test 1. OLS to get the residuals e t 2. e t = γ 1 e t γ p e t p + x tβ + δ t 3. H 0 = n R 2 χ 2 p 11

14 EViews Session II: Estimating a money demand equation What to do in case of 1) Heteroscedasticity, 2) Autocorrelation? GLS (if Ω known), or FGLS OLS with 1) White VC matrix, or 2) Newey - West VC matrix Empirical Example: Money demand. Transaction demand GDP Speculation Demand Bond rate, Money market rate 12

15 EViews Session II: Estimating a money demand equation a) Open the workfile MONEY.WF1. You find the following variables: Variable bondr gdp m3 monrat Description bondrate GDP (real, seasonally adjusted) M3 (seasonally adjusted) money market interest rate 13

16 EViews Session II: Estimating a money demand equation b) Generate the following variables... l_gdp = log(gdp), l_m3 = log(m3). c)... and estimate the following equation (you take into account the transaction demand): l_m3 = β 0 + β 1 l_gdp + ε. Save it as eq1. d) Has ˆβ 1 the sign you expected? Interpret. 14

17 EViews Session II: Estimating a money demand equation e) Now you consider the speculation demand as well and estimate the following equation: l_m3 = β 0 + β 1 l_gdp + β 2 monrat + β 3 bondr + ε. Save it as eq2. f) Have the coefficients the signs you expected? How do you interpret β 1, β 2 and β 3? g) Now look at the residual plot. (View -> Actual, Fitted, Residual -> Residual Graph). Do you find evidence of the presence of heteroscedasticity and/or autocorrelation? 15

18 EViews Session II: Estimating a money demand equation h) Consider the Durbin-Watson statistic: 1. Is there evidence of autocorrelation? If yes, positive or negative correlation? 2. Which assumptions does an application of the Durbin-Watson test require? 3. Are these assumptions valid in this case? i) Run a Breusch-Godfreyserial correlation test including 4 lags. 1. What is the null hypothesis? 2. Procedure in EViews: View -> Residual Tests -> Serial Correlation LM Test Do you reject the null hypothesis on a 5% significance level? 16

19 EViews Session II: Estimating a money demand equation j) Estimate the second equation, now using the Newey-West-VC-Matrix (click on Options -> Heteroscedasticity -> Newey-West in the specification window). Save the result as eq3. How do the t-values change compared to equation 2? k) Based on equation 3, test the following hypothesis on a 5% significance level: i) β 2 = 0, ii) β 1 = 2 and iii) β 3 < 0. Address the following points: 1. What are the null and the alternative hypothesis? 2. Calculate an appropriate test statistic. 3. Find the 5% critical value. 4. Do you reject the null hypothesis? 17

20 EViews Session II: Estimating a money demand equation l) Test if the additional parameters in equation 2 (and equation 3) are jointly significant. Calculate the appropriate F statistic using two different ways: 1. Use the F test that is implemented in EViews. 2. Calculate the F test by hand using both regression outputs. 18

21 EViews Session III: Monte Carlo Simulation A typical Monte Carlo simulation exercise consists of the following steps: 1. Specify the true model (data generating process) underlying the data. 2. Simulate a draw from the data and estimate the model using the simulated data. 3. Repeat step 2 many times, each time storing the results of interest. 4. The end result is a series of estimation results, one for each repetition of step 2. We can then characterize the empirical distribution of these results by tabulating the sample moments or by plotting the histogram or kernel density estimate. 19

22 EViews Session III: Monte Carlo Simulation Step 2 typically involves simulating a random draw from a specified distribution. EViews provides built-in pseudo-random number generating functions for a wide range of commonly used distributions; The step that requires a little thinking is how to store the results from each repetition (step 3). There are two methods that you can use in EViews: 1. Storing results in a series; 2. Storing results in a matrix. 20

23 EViews Session III: Monte Carlo Simulation 1) Storing results in a series (or group of series): The difficulty with this approach is that series elements are most easily indexed by a sample in EViews. To store the result from each replication as a different observation in a series, you must shift the sample every time you store a new result. Moreover, the length of the series will be constrained by the size of your workfile sample: for example, if you wish to perform 1000 replications on a workfile with 100 observations, you will not be able to store all 1000 results in a series since the latter only has 100 observations. 21

24 EViews Session III: Monte Carlo Simulation 2) Storing results in a matrix (or vector): The matrix is indexed by its row-column position and its size is independent of the workfile sample. For example, you can declare a matrix with 1000 rows and 10 columns in a workfile with only 1 observation. The disadvantage of the matrix method is that the matrix object does not have as much built-in functions as a series object. For example, there is no kernel density estimate view out of a matrix (which is available for a series object). For didactic purposes, I will illustrate both approaches. (My own recommendation is a mixed approach. Store all the results in a matrix. If you need to do further processing, convert the matrix into a group of series.) 22

25 EViews Session III: Monte Carlo Simulation As a concrete example, consider the following exercise (3.26) from Damodar Gujarati Basic Econometrics, 3rd edition: Refer to the 10 X values of Table 3.2 (which are: 80, 100, 120, 140, 160, 180, 200, 220, 240, 260). Let Beta(1) = 2.5 and Beta(2) = 0.5. Assume that the errors are distributed N(0, 9), that is, the errors are normally distributed with mean 0 and variance 9. Generate 100 samples using these values, obtaining 100 estimates of Beta(1) and Beta(2). Graph these estimates. What conclusions can you draw from the Monte Carlo study? 23

26 EViews Session III: Monte Carlo Simulation 1) Store monte carlo results in a series set workfile range to number of monte carlo replications wfcreate mcarlo u create data series for x NOTE: x is fixed in repeated samples only first 10 observations are used (remaining 90 obs missing) series x x.fill 80, 100, 120, 140, 160, 180, 200, 220, 240, 260 set true parameter values!beta1 = 2.5!beta2 = 0.5 set seed for random number generator rndseed assign number of replications to a control variable!reps =

27 EViews Session III: Monte Carlo Simulation begin loop for!i = 1 to!reps set sample to estimation sample smpl 1 10 simulate y data (only for 10 obs) series y =!beta1 +!beta2*x + 3*nrnd regress y on a constant and x equation eq1.ls y c x set sample to one observation smpl!i!i and store each coefficient estimate in a series series b1 = eq1.@coefs(1) series b2 = eq1.@coefs(2) next end of loop 25

28 EViews Session III: Monte Carlo Simulation set sample to full sample smpl show kernel density estimate for each coef freeze(gra1) b1.kdensity draw vertical dashline at true parameter value gra1.draw(dashline, bottom, rgb(156,156,156))!beta1 show gra1 freeze(gra2) b2.kdensity draw vertical dashline at true parameter value gra2.draw(dashline, bottom, rgb(156,156,156))!beta2 show gra2 26

29 EViews Session III: Monte Carlo Simulation 2) Store monte carlo results in a Matrix set workfile range to number of obs wfcreate mcarlo u 1 10 create data series for x NOTE: x is fixed in repeated samples series x x.fill 80, 100,120, 140, 160, 180, 200, 220, 240, 260 set seed for random number generator rndseed assign number of replications to a control variable!reps = 100 declare storage matrix matrix(!reps,2) beta 27

30 EViews Session III: Monte Carlo Simulation begin loop for!i = 1 to!reps simulate y data series y = *x + 3*nrnd regress y on a constant and x equation eq1.ls y c x store each coefficient estimate in matrix beta(!i,1) = eq1.@coefs(1) column 1 is intercept beta(!i,2) = eq1.@coefs(2) column 2 is slope next end of loop show descriptive stats of coef distribution beta.stats 28