1 A MONTE CARLO EXPERIMENT In the previous slideshow, we saw that the error term is responsible for the variations of b 2 around its fixed component 

Slides:

Advertisements

Similar presentations

1 MAXIMUM LIKELIHOOD ESTIMATION OF REGRESSION COEFFICIENTS X Y XiXi 11  1  +  2 X i Y =  1  +  2 X We will now apply the maximum likelihood principle.

Advertisements

EC220 - Introduction to econometrics (chapter 3)

Christopher Dougherty EC220 - Introduction to econometrics (chapter 2) Slideshow: a Monte Carlo experiment Original citation: Dougherty, C. (2012) EC220.

Christopher Dougherty EC220 - Introduction to econometrics (chapter 10) Slideshow: introduction to maximum likelihood estimation Original citation: Dougherty,

1 THE DISTURBANCE TERM IN LOGARITHMIC MODELS Thus far, nothing has been said about the disturbance term in nonlinear regression models.

EC220 - Introduction to econometrics (chapter 7)

1 XX X1X1 XX X Random variable X with unknown population mean  X function of X probability density Sample of n observations X 1, X 2,..., X n : potential.

1 THE NORMAL DISTRIBUTION In the analysis so far, we have discussed the mean and the variance of a distribution of a random variable, but we have not said.

HETEROSCEDASTICITY-CONSISTENT STANDARD ERRORS 1 Heteroscedasticity causes OLS standard errors to be biased is finite samples. However it can be demonstrated.

1 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE red This sequence provides an example of a discrete random variable. Suppose that you.

Random effects estimation RANDOM EFFECTS REGRESSIONS When the observed variables of interest are constant for each individual, a fixed effects regression.

MEASUREMENT ERROR 1 In this sequence we will investigate the consequences of measurement errors in the variables in a regression model. To keep the analysis.

ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY

1 ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA Assumptions C.1, C.3, C.4, C.5, and C.8, and the consequences of their violations are the.

EC220 - Introduction to econometrics (chapter 2)

EC220 - Introduction to econometrics (chapter 9)

EXPECTED VALUE OF A RANDOM VARIABLE 1 The expected value of a random variable, also known as its population mean, is the weighted average of its possible.

TESTING A HYPOTHESIS RELATING TO THE POPULATION MEAN 1 This sequence describes the testing of a hypothesis at the 5% and 1% significance levels. It also.

Christopher Dougherty EC220 - Introduction to econometrics (chapter 3) Slideshow: prediction Original citation: Dougherty, C. (2012) EC220 - Introduction.

1 THE CENTRAL LIMIT THEOREM If a random variable X has a normal distribution, its sample mean X will also have a normal distribution. This fact is useful.

Cross-sectional:Observations on individuals, households, enterprises, countries, etc at one moment in time (Chapters 1–10, Models A and B). 1 During this.

Christopher Dougherty EC220 - Introduction to econometrics (chapter 10) Slideshow: maximum likelihood estimation of regression coefficients Original citation:

DERIVING LINEAR REGRESSION COEFFICIENTS

Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: the normal distribution Original citation: Dougherty, C. (2012)

1 In a second variation, we shall consider the model shown above. x is the rate of growth of productivity, assumed to be exogenous. w is now hypothesized.

1 This sequence shows why OLS is likely to yield inconsistent estimates in models composed of two or more simultaneous relationships. SIMULTANEOUS EQUATIONS.

1 PREDICTION In the previous sequence, we saw how to predict the price of a good or asset given the composition of its characteristics. In this sequence,

EC220 - Introduction to econometrics (review chapter)

1 UNBIASEDNESS AND EFFICIENCY Much of the analysis in this course will be concerned with three properties of estimators: unbiasedness, efficiency, and.

FIXED EFFECTS REGRESSIONS: WITHIN-GROUPS METHOD The two main approaches to the fitting of models using panel data are known, for reasons that will be explained.

Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: sampling and estimators Original citation: Dougherty, C. (2012)

Christopher Dougherty EC220 - Introduction to econometrics (chapter 12) Slideshow: autocorrelation, partial adjustment, and adaptive expectations Original.

THE DUMMY VARIABLE TRAP 1 Suppose that you have a regression model with Y depending on a set of ordinary variables X 2,..., X k and a qualitative variable.

Christopher Dougherty EC220 - Introduction to econometrics (chapter 8) Slideshow: measurement error Original citation: Dougherty, C. (2012) EC220 - Introduction.

THE FIXED AND RANDOM COMPONENTS OF A RANDOM VARIABLE 1 In this short sequence we shall decompose a random variable X into its fixed and random components.

CONSEQUENCES OF AUTOCORRELATION

ALTERNATIVE EXPRESSION FOR POPULATION VARIANCE 1 This sequence derives an alternative expression for the population variance of a random variable. It provides.

CONFLICTS BETWEEN UNBIASEDNESS AND MINIMUM VARIANCE

1 t TEST OF A HYPOTHESIS RELATING TO A POPULATION MEAN The diagram summarizes the procedure for performing a 5% significance test on the slope coefficient.

ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR

EC220 - Introduction to econometrics (chapter 8)

MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS

MULTIPLE REGRESSION WITH TWO EXPLANATORY VARIABLES: EXAMPLE 1 This sequence provides a geometrical interpretation of a multiple regression model with two.

Christopher Dougherty EC220 - Introduction to econometrics (chapter 12) Slideshow: footnote: the Cochrane-Orcutt iterative process Original citation: Dougherty,

Simple regression model: Y =  1 +  2 X + u 1 We have seen that the regression coefficients b 1 and b 2 are random variables. They provide point estimates.

A.1The model is linear in parameters and correctly specified. PROPERTIES OF THE MULTIPLE REGRESSION COEFFICIENTS 1 Moving from the simple to the multiple.

Christopher Dougherty EC220 - Introduction to econometrics (chapter 9) Slideshow: instrumental variable estimation: variation Original citation: Dougherty,

Christopher Dougherty EC220 - Introduction to econometrics (chapter 6) Slideshow: multiple restrictions and zero restrictions Original citation: Dougherty,

1 We will now look at the properties of the OLS regression estimators with the assumptions of Model B. We will do this within the context of the simple.

1 Y SIMPLE REGRESSION MODEL Suppose that a variable Y is a linear function of another variable X, with unknown parameters  1 and  2 that we wish to estimate.

1 We will continue with a variation on the basic model. We will now hypothesize that p is a function of m, the rate of growth of the money supply, as well.

1 ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION In practice we deal with finite samples, not infinite ones. So why should we be interested.

Definition of, the expected value of a function of X : 1 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE To find the expected value of a function of.

HETEROSCEDASTICITY 1 This sequence relates to Assumption A.4 of the regression model assumptions and introduces the topic of heteroscedasticity. This relates.

INSTRUMENTAL VARIABLES 1 Suppose that you have a model in which Y is determined by X but you have reason to believe that Assumption B.7 is invalid and.

1 INSTRUMENTAL VARIABLE ESTIMATION OF SIMULTANEOUS EQUATIONS In the previous sequence it was asserted that the reduced form equations have two important.

1 HETEROSCEDASTICITY: WEIGHTED AND LOGARITHMIC REGRESSIONS This sequence presents two methods for dealing with the problem of heteroscedasticity. We will.

1 ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION We have seen that the variance of a random variable X is given by the expression above. Variance.

1 REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION Linear restrictions can also be tested using a t test. This involves the reparameterization.

1 We will illustrate the heteroscedasticity theory with a Monte Carlo simulation. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION 1 standard deviation of.

Christopher Dougherty EC220 - Introduction to econometrics (chapter 1) Slideshow: simple regression model Original citation: Dougherty, C. (2012) EC220.

FOOTNOTE: THE COCHRANE–ORCUTT ITERATIVE PROCESS 1 We saw in the previous sequence that AR(1) autocorrelation could be eliminated by a simple manipulation.

VARIABLE MISSPECIFICATION I: OMISSION OF A RELEVANT VARIABLE In this sequence and the next we will investigate the consequences of misspecifying the regression.

Introduction to Econometrics, 5th edition

Introduction to Econometrics, 5th edition

Introduction to Econometrics, 5th edition

Introduction to Econometrics, 5th edition

Introduction to Econometrics, 5th edition

Introduction to Econometrics, 5th edition

Introduction to Econometrics, 5th edition

Presentation transcript:

1 A MONTE CARLO EXPERIMENT In the previous slideshow, we saw that the error term is responsible for the variations of b 2 around its fixed component  2. We demonstrated mathematically that the expectation of the error term is zero and hence that b 2 is an unbiased estimator of  2. True model Fitted model

2 In this slideshow we will investigate the effect of the error term on b 2 directly, using a Monte Carlo experiment (simulation). A MONTE CARLO EXPERIMENT True model Fitted model

3 A Monte Carlo experiment is a laboratory-style exercise usually undertaken with the objective of evaluating the properties of regression estimators under controlled conditions. A MONTE CARLO EXPERIMENT Choose model in which Y is determined by X, parameter values, and u

4 We will use one to investigate the behavior of OLS regression coefficients when applied to a simple regression model. A MONTE CARLO EXPERIMENT Choose model in which Y is determined by X, parameter values, and u

5 We will assume that Y is determined by a variable X and a disturbance term u, we will choose the data for X, and we will choose values for the parameters. A MONTE CARLO EXPERIMENT Choose data for X Choose parameter values Choose model in which Y is determined by X, parameter values, and u

6 We will also generate values for the disturbance term randomly from a known distribution. A MONTE CARLO EXPERIMENT Choose data for X Choose parameter values Choose distribution for u Choose model in which Y is determined by X, parameter values, and u

7 The values of Y in the sample will be determined by the values of X, the parameters and the values of the disturbance term. A MONTE CARLO EXPERIMENT Choose data for X Choose parameter values Choose distribution for u Model Generate the values of Y Choose model in which Y is determined by X, parameter values, and u

8 We will then use the regression technique to obtain estimates of the parameters using only the data on Y and X. A MONTE CARLO EXPERIMENT Choose data for X Choose parameter values Choose distribution for u Model Generate the values of Y Estimators Estimate the values of the parameters Choose model in which Y is determined by X, parameter values, and u

9 We can repeat the process indefinitely keeping the same data for X and the same values of the parameters but using new randomly-generated values for the disturbance term. A MONTE CARLO EXPERIMENT Choose data for X Choose parameter values Choose distribution for u Model Generate the values of Y Estimators Estimate the values of the parameters Choose model in which Y is determined by X, parameter values, and u

10 In this way we can derive probability distributions for the regression estimators which allow us, for example, to check up on whether they are biased or unbiased. A MONTE CARLO EXPERIMENT Choose data for X Choose parameter values Choose distribution for u Model Generate the values of Y Estimators Estimate the values of the parameters Choose model in which Y is determined by X, parameter values, and u

11 In this experiment we have 20 observations in the sample. X takes the values 1, 2,..., 20.  1 is equal to 2.0 and  2 is equal to 0.5. A MONTE CARLO EXPERIMENT Choose data for X Choose parameter values Choose distribution for u Model Generate the values of Y Estimators Estimate the values of the parameters Y =  1 +  2 X + u X = 1, 2,..., 20  1 = 2.0  2 = 0.5 u is independent N(0,1) Y = X + u Generate the values of Y Choose model in which Y is determined by X, parameter values, and u

12 The disturbance term is generated randomly using a normal distribution with zero mean and unit variance. Hence we generate the values of Y. A MONTE CARLO EXPERIMENT Choose data for X Choose parameter values Choose distribution for u Model Generate the values of Y Estimators Estimate the values of the parameters Y =  1 +  2 X + u X = 1, 2,..., 20  1 = 2.0  2 = 0.5 u is independent N(0,1) Y = X + u Generate the values of Y Choose model in which Y is determined by X, parameter values, and u

13 We will then regress Y on X using the OLS estimation technique and see how well our estimates b 1 and b 2 correspond to the true values  1 and  2. Choose data for X Choose parameter values Choose distribution for u Model Generate the values of Y Estimators Estimate the values of the parameters Y =  1 +  2 X + u X = 1, 2,..., 20  1 = 2.0  2 = 0.5 u is independent N(0,1) Y = X + u Generate the values of Y Estimate the values of the parameters A MONTE CARLO EXPERIMENT Choose model in which Y is determined by X, parameter values, and u

Here are the values of X, chosen quite arbitrarily. 14 A MONTE CARLO EXPERIMENT X X u YX X u Y 12.5– – – – – – – – – – – Y = X + u

Given our choice of numbers for  1 and  2, we can derive the nonstochastic component of Y. 15 A MONTE CARLO EXPERIMENT X X u YX X u Y 12.5– – – – – – – – – – – Y = X + u

The nonstochastic component is displayed graphically. 16 A MONTE CARLO EXPERIMENT Y = X + u

Next, we generate randomly a value of the disturbance term for each observation using a N(0,1) distribution (normal with zero mean and unit variance). 17 A MONTE CARLO EXPERIMENT X X u YX X u Y 12.5– – – – – – – – – – – Y = X + u

18 A MONTE CARLO EXPERIMENT X X u YX X u Y 12.5– – – – – – – – – – – Thus, for example, the value of Y in the first observation is 1.91, not Y = X + u

19 Similarly, we generate values of Y for the other 19 observations. A MONTE CARLO EXPERIMENT X X u YX X u Y 12.5– – – – – – – – – – – Y = X + u

The 20 observations are displayed graphically. 20 A MONTE CARLO EXPERIMENT Y = X + u

We have reached this point in the Monte Carlo experiment. 21 A MONTE CARLO EXPERIMENT Choose data for X Choose parameter values Choose distribution for u Model Generate the values of Y Estimators Estimate the values of the parameters Y =  1 +  2 X + u X = 1, 2,..., 20  1 = 2.0  2 = 0.5 u is independent N(0,1) Y = X + u Generate the values of Y A MONTE CARLO EXPERIMENT Choose model in which Y is determined by X, parameter values, and u

We will now apply the OLS estimators for b 1 and b 2 to the data for X and Y, and see how well the estimates correspond to the true values. 22 Choose data for X Choose parameter values Choose distribution for u Model Generate the values of Y Estimators Estimate the values of the parameters Y =  1 +  2 X + u X = 1, 2,..., 20  1 = 2.0  2 = 0.5 u is independent N(0,1) Y = X + u Generate the values of Y Estimate the values of the parameters A MONTE CARLO EXPERIMENT Choose model in which Y is determined by X, parameter values, and u

Here is the scatter diagram again. 23 A MONTE CARLO EXPERIMENT Y = X + u

The regression estimators use only the observed data for X and Y. 24 A MONTE CARLO EXPERIMENT Y = X + u

Here is the regression equation fitted to the data. 25 A MONTE CARLO EXPERIMENT Y = X + u

For comparison, the nonstochastic component of the true relationship is also displayed.  2 (true value 0.50) has been overestimated and  1 (true value 2.00) has been underestimated. 26 A MONTE CARLO EXPERIMENT Y = X + u

We will repeat the process, starting with the same nonstochastic components of Y. 27 A MONTE CARLO EXPERIMENT Y = X + u

As before, the values of Y are modified by adding randomly-generated values of the disturbance term. 28 A MONTE CARLO EXPERIMENT Y = X + u

The new values of the disturbance term are drawn from the same N(0,1) distribution as the previous ones but, except by coincidence, will be different from them. 29 A MONTE CARLO EXPERIMENT Y = X + u

This time the slope coefficient has been underestimated and the intercept overestimated. 30 A MONTE CARLO EXPERIMENT Y = X + u

We will repeat the process once more. 31 A MONTE CARLO EXPERIMENT Y = X + u

A new set of random numbers has been used in generating the values of Y. 32 A MONTE CARLO EXPERIMENT Y = X + u

As last time, the slope coefficient has been underestimated and the intercept overestimated. 33 A MONTE CARLO EXPERIMENT Y = X + u

The table summarizes the results of the three regressions and adds those obtained repeating the process a further seven times. 34 sample b 1 b A MONTE CARLO EXPERIMENT

Here is a histogram for the estimates of  2. Nothing much can be seen yet. 35 A MONTE CARLO EXPERIMENT 10 samples

Here are the estimates of  2 obtained with 40 further replications of the process A MONTE CARLO EXPERIMENT

The histogram is beginning to display a central tendency. 37 A MONTE CARLO EXPERIMENT 50 samples

This is the histogram with 100 replications. We can see that the distribution appears to be symmetrical around the true value, implying that the estimator is unbiased. 38 A MONTE CARLO EXPERIMENT 100 samples

However, the distribution is still rather jagged. It would be better to repeat the process 1,000,000 times, perhaps more. 39 A MONTE CARLO EXPERIMENT 100 samples

The red curve shows the limiting shape of the distribution. It is symmetrical around the true value, indicating that the estimator is unbiased. 40 A MONTE CARLO EXPERIMENT 100 samples

The distribution is normal because the random component of the coefficient is a weighted linear combination of the values of the disturbance term in the observations in the sample. We demonstrated this in the previous slideshow. 41 A MONTE CARLO EXPERIMENT 100 samples

We are assuming (Assumption A.6) that the disturbance term in each observation has a normal distribution. A linear combination of normally distributed random variables itself has a normal distribution. 42 A MONTE CARLO EXPERIMENT 100 samples

Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 2.4 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course EC2020 Elements of Econometrics