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

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1 ECONOMETRIC MODELS The concept of Data Generating Process (DGP) and its relationships with the analysis of specification. 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 = β 0 + β 1 y t 1 + u t u t WN(0,σ 2 u) t =1, 2,...,T y 0 =0. Second, set the parameter values β 0 =0.5,β 1 =0.7, σ 2 u =0.5. Third, specify a stochastic distribution for the variables: u t WNGaussian(0,σ 2 u) WNN(0,σ 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

5 4 y(t) = *y(t-1) + u(t), u(t) drawn from N(0, 0.5) T = 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 specifies 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 := 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 y 0 ; δ 0,δ 1,σ 2 u):=f(y T y T 1,...,y 0 ) f(y T 1 y T 2,...,y 0 )... f(y 1 y 0 ) := TY t=1 f(y t y 0, F t 1 ), F t 1 := {y t 1,...,y 1 }.

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 = δ 0 +δ 1 y t 1 +u t, u t WNN(0,σ 2 u), t =1,...,T whose unknown parameters are δ 0,δ 1, σ 2 u.

9 The unknown parameters δ 0,δ 1, σ 2 u can be inferred from the data y 0,y 1,y 2,...,y T by estimating the specified statistical model under the maintained assumption that the DGP belongs to the specified 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 specified!

10 We say that a statistical model is correctly specified if it captures salient aspects of the DGP: Extremely good case: the DGP belongs to the specified statistical model (it means that the DGP is obtained from the statistical model by fixing the unknown parameters to their true value). In this case the statistician/econometrician recovers consistent estimates of the unknown parameters and, possibly, efficients, 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 (difficult to say something about efficiency). Correct inference on the unknown parameters. In this case, the distribution specified in the statistical model and/or the sampling scheme may differ from the true distribution and sampling scheme in the DGP, but the extent of such difference does not affect the possibility of estimating the parameters consistently.

11 Recall Section Deterministic sequences Let {h T,T =1, 2,...} {h T } be a sequence of real numbers. If the sequence has a limit, h, then this is denoted by lim h T = h. T This implies that for every ε>0 there exists a positive, finite integer T ε such that h T h <ε for T>T ε. If h T is a p 1 vector, lim T h T = h means that for every ε>0 there exists a positive, finite integer T ε such that kh T hk 2 <ε for T>T ε.

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 p 1. One can generalize this measure by defining the norm kvk A := (v 0 Av) 1/2 where A is a symmetric positive definite matrix; this norm measures the distance of v from 0 p 1 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 {h T,T =1, 2,...} converges in probability to the non-stochastic vector h if for all >0: lim P (kh T hk T 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 ˆθ T be the estimator of the unknown parameter θ 0 obtained from a sample of length T, and consider the sequence nˆθ T,T =1, 2,... o (hence random vectors); then ˆθ T is said to be a consistent estimator of θ 0 if ˆθ T p θ 0. Convergence in probability implies that the difference between ˆθ T and θ 0 disappears with probability one as T. In the limit ˆθ T and θ 0 are essentially identical. End of Recall Section

15 Example 1. The DGP is as above and the statistician/econometrician specifies y t = β 0 + β 1 y t 1 + β 2 z t + u t, u t WNN(0,σ 2 u) where z t is iid and is irrelevant with respect to the DGP. He/she can still get consistent estimates of β 0,β 1 and σ 2 u basedontheonbervatios 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 specified, i.e. is misspecified, if it provides inconsistent estimates of the unknown parameters. Example 2. DGP as above but the statistician/econometrician specifies: y t = β 0 + u t, u t WNN(0,σ 2 u). The OLS (ML) estimators of β 0 σ 2 u based on are not consistent! y 0,y 1,y 2,...,y T

17 Example 3 (structural break) DGP: y t = y 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 specifies the statistical model: y t = β 0 + β 1 y t 1 + u t, u t WNN(0,σ 2 u). Here the OLS (or ML) estimator of β 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, andletw t the log of real per-capita financial wealth of the agent at time t. Imagine that according to the chosen theory: c t = β 0 +β 1 c t 1 +β 2 w t 1 +u t, u t WN(0,σ 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 financial 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 = β 0 +β 1 c t 1 +β 2 w t +u t, u t WN(0,σ 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 disposableincomeamongdifferent financial assets) might be simultaneous. Since portfolio decisions at time t affect w t, it follows that the knowlegde of w t might require the contemporaneous knowledge of c t!

21 The predeterminate variables, by definition, do not contain also endogenous variables, i.e. variables that the model attempts to explain or that are directly influenced at time t bythevariablethemodelattempts to expalin. A correctly specified econometric model in reduced form should not be affected 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 = ρr t 1 +(1 ρ)[ϕ b b t + ϕ y π t ]+u t b t = α 1 b t 1 α 2 b t 2 δ(r t π t )+η t! ÃÃ! " 0 σ 2 WNN, u σ u,η 0 Ã ut η t σ 2 η #! Reduced Form: R t = π 11 R t 1 + π 12 b t 1 +π 13 b t 2 + π 14 π t + ε R t b t = π 21 R t 1 + π 22 b t 1 + π 23 b t 2 ++π 24 π t + ε b t Ã! ÃÃ! " #! ε R t 0 σ 2 ε b WNN, 1 σ 1,2 t 0 σ 2. 2 Obvioulsy, the parameters of the two systems are strictly (linearly) connected.

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 specified (because the DGP is unknown by definition). 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.