ECONOMETRIC MODELS. The concept of Data Generating Process (DGP) and its relationships with the analysis of specication.

Transcription

1 ECONOMETRIC MODELS The concept of Data Generating Process (DGP) and its relationships with the analysis of specication. Luca Fanelli University of Bologna

2 The concept of Data Generating Process (DGP) A convenient way to understand the concept of DGP is to imagine to perform a simulation experiment. Monte Carlo experiment. Exercise: Simulate data from a (scalar) stationary AR(1) model.

3 Exercise: Simulate data from a (scalar) stationary AR(1) model. First, set the model: y t = y t 1 + u t u t WN(0, 2 u) t = 1; 2; :::; T y 0 = 0: Second, set the parameter values 0 = 0:5; 1 = 0:7; 2 u = 0:5: Third, specify a stochastic distribution for the variables: u t WNGaussian(0, 2 u) WNN(0, 2 u)

4 We know exactly the stochastic distribution of y t because we are using it to generate the data! We know exactly the mechanism through which the sequence of T observations (given y 0 ) is generated. y 1 ; y 2 ; :::; y T If you use R or any other statistical/econometric package you can run the above experiment yourself.

5 A possible realization of T=300 observations from the DGP

6 DGP and statistical model In the Monte Carlo experiment above we have simulated a simple economy. In real cases, the investigator does not know how the sequence y 0 ; y 1 ; y 2 ; :::; y T has been generated. He/she species a statistical model which attempts to approximate the `true' DGP (at least its salient features) as best as possible.

7 What is a statistical model? Statistical Model := 8 >< >: stochastic distribution + sampling scheme The statistical model is called parametric when the only unknown quantity are the parameters that characterize the stochastic distribution. Given a parametric statistical model, one can always write down the joint distribution of the observations (data) by using the sequential factorization: f(y 1 ; y 2 ; :::; y T j y 0 ; 0 ; 1 ; 2 u):=f(y T j y T 1 ; :::; y 0 ) f(y T 1 j y T 2 ; :::; y 0 ) ::: f(y 1 j y 0 ) := TY t=1 f(y t j y 0, F t 1 ), F t 1 := fy t 1 ; :::; y 1 g :

8 The joint distribution that summarizes the two crucial ingredients of a statistical model is known as likelihood function (a part from a constant). Recall: when you write a likelihood function you have an underlying statististical model! As an example, assume that the statistician/econometrician deems that given y 0, the sequence y 1 ; y 2 ; :::; y T is generated by the following statistical (parametric) model: y t = y t 1 +u t, u t WNN(0, 2 u), t = 1; :::; T whose unknown parameters are 0 ; 1, 2 u:

9 The unknown parameters 0 ; 1, 2 u can be inferred from the data y 0 ; y 1 ; y 2 ; :::; y T by estimating the specied statistical model under the maintained assumption that the DGP belongs to the specied statistical model (i.e. under the postulated stochastic distribution and sampling scheme). The likelihood function allows the statistician to recover ML estimates of the unknown parameters. In general, we like ML estimation because of its `nice' properties when the model is correctly specied!

10 We say that a statistical model is correctly specied if it captures salient aspects of the DGP: Extremely good case: the DGP belongs to the specied statistical model (it means that the DGP is obtained from the statistical model by xing the unknown parameters to their `true' value). In this case the statistician/econometrician recovers consistent estimates of the unknown parameters and, possibly, ef- cients, i.e. with minimum variance; Reasonably good case: the estimation of the statistical model allows the statistician/econometrician to recover consistent estimates of the unknown parameters (dicult to say something about eciency). )Correct inference on the unknown parameters. In this case, the distribution specied in the statistical model and/or the sampling scheme may dier from the `true' distribution and sampling scheme in the DGP, but the extent of such dierence does not aect the possibility of estimating the parameters consistently.

11 Recall Section Deterministic sequences Let fh T ; T = 1; 2; :::g fh T g be a sequence of real numbers. If the sequence has a limit, h, then this is denoted by lim T!1 h T = h: This implies that for every " > 0 there exists a positive, nite integer T " such that jh T hj < " for T > T " : If h T is a p 1 vector, lim T!1 h T = h means that for every " > 0 there exists a positive, nite integer T " such that kh T hk 2 < " for T > T " :

12 Note that kvk 2 :=(v 0 v) 1=2 is the Euclidean norm of the vector v. This can be interpreted as a measure of the length of v in the space R p, i.e. a measure of the distance of the vector v from the vector 0 p1. One can generalize this measure by dening the norm kvk A := (v 0 Av) 1=2 where A is a symmetric positive denite matrix; this norm measures the distance of v from 0 p1 `weighted' by the elements of the matrix A.

13 Stochastic sequences Henceforth h T will be considered a p 1 vector, except where stated otherwise Suppose now that each h T is a (continuous) random vector. We are interested in the concepts of convergence in probability and convergence in distribution. The sequence of random vectors fh T ; T = 1; 2; :::g converges in probability to the non-stochastic vector h if for all > 0: lim P (kh T hk T!1 2 < ) = 1; we conventionally write h T! p h:

14 The concept of convergence in probability leads us to the concept of consistency of an estimator. Consistency of an estimator Let ^T be the estimator of the unknown parameter 0 obtained from a sample of length T, and consider the sequence n^t ; T = 1; 2; ::: o (hence random vectors); then ^T is said to be a consistent estimator of 0 if ^T! p 0. Convergence in probability implies that the dierence between ^T and 0 disappears with probability one as T! 1: In the limit ^T and 0 are essentially identical. End of Recall Section

15 Example 1. The DGP is as above and the statistician/econometrician species y t = y t z t + u t, u t WNN(0, 2 u) where z t is iid and is irrelevant with respect to the DGP. He/she can still get consistent estimates of 0 ; 1 and 2 u based on the onbervatios y 0 ; y 1 ; y 2 ; :::; y T z 1 ; z 2 ; :::; z T :

16 In turn, we say that a statistical model is not correctly specied, i.e. is misspecied, if it provides inconsistent estimates of the unknown parameters. Example 2. DGP as above but the statistician/econometrician species: y t = 0 + u t, u t WNN(0, 2 u ). The OLS (ML) estimators of 0 2 u based on are not consistent! y 0 ; y 1 ; y 2 ; :::; y T

17 Example 3 (structural break) DGP: y t = 0:5 + 0:7y t 1 (1 D t ) 0:3y t 1 D t + u t u t WNN(0,1) dummy: D t = ( 1 if t T1 0 otherwise, 1 T 1 T Econometrician/statisticain species the statistical model: y t = y t 1 + u t, u t WNN(0, 2 u). Here the OLS (or ML) estimator of 1 is not consistent!

18 DGP, statistical model and econometric model Which relationship exists between the statistical model and the (dynamic) econometric model? Econometricians usually call `statistical model' what in their jargon is an econometric model in `reduced form'. An econometric model can usually be expressed in two forms: reduced form and structural form: Econometric model = ( reduced form representation structural form representation.

19 An econometric model in reduced form is a model in which the endogenous variable(s) at time t depend only on a set of variables, called predeterminated variables, such that in order to know this set of variables at time t one does need to know the value of the endogenous variable at time t. Example 4. We want to explain the consumption behaviour of an economic agent. Let c t be the log real per-capita consumption of the agent at time t, and let w t the log of real per-capita nancial wealth of the agent at time t. Imagine that according to the chosen theory: c t = c t w t 1 +u t, u t WN(0, 2 u), t = 1; :::; T: In this example, c t is the endogenous variable and x t :=(1; c t 1 ; w t 1 ) 0 the vector of predeterminated variables. According to this model, the consumption level of the agent at time t depends on a constant, the consumption level in the previous period (habit persistence) and the level of nancial wealth in the previous period; the knowledge of each element of x t does not require the knowledge of c t!

20 Example 4 (continued). Imagine now that the theory instead predicts that c t = c t w t +u t, u t WN(0, 2 u) t = 1; :::; T: Is the vector x t :=(1; c t 1 ; w t ) 0 still predetermined? We have the following doubt. Consumption and portfolio decisions (the allocation of non-consumed disposable income among dierent nancial assets) might be simultaneous. Since portfolio decisions at time t aect w t, it follows that the knowlegde of w t might require the contemporaneous knowledge of c t!

21 The predeterminate variables, by denition, do not contain also endogenous variables, i.e. variables that the model attempts to explain or that are directly inuenced at time t by the variable the model attempts to expalin. A correctly specied econometric model in reduced form should not be aected by the so-called endogeneity bias issue. Thus the econometric model coincides with the statistical model when it is expressed in reduced form.

22 Example 5. Structural Form: R t = R t 1 + (1 )[' b b t + ' y t ] + u t b t = 1 b t 1 2 b t 2 (R t t ) + t u t t! W NN 0 0!, " 2 u u; 2 #! Reduced Form: R t = 11 R t b t b t t + " R t b t = 21 R t b t b t t + " b t " R t " b t! W NN 0 0!, " 2 1 1;2 Obvioulsy, the parameters of the two systems are strictly (linearly) connected. 2 2 #! :

23 When we specify an econometric model, our ambition is that its reduced form (statistical model) approximates as close as possible the features of the underlying DGP. Of course, an investigator will never know that its statistical model is correctly specied (because the DGP is unknown by denition). Imagine that the economy (or the market) has done a Monte Carlo simulation and generated some observations. The econometrician/statisticain does not know the actual features of the experiment. However, he/she uses his/her theoretical knowledge about the phenomenon of interest and the available data to infer the salient feature of that experiment.