Graduate Econometrics I: What is econometrics?

Transcription

1 Graduate Econometrics I: What is econometrics? Yves Dominicy Université libre de Bruxelles Solvay Brussels School of Economics and Management ECARES Yves Dominicy Graduate Econometrics I: What is econometrics? 1/23

2 Outline 1 2 Yves Dominicy Graduate Econometrics I: What is econometrics? 2/23

3 Outline 1 2 Yves Dominicy Graduate Econometrics I: What is econometrics? 3/23

4 Almost everything that we are going to see and study in this course goes around three key words : Model Information Risk Yves Dominicy Graduate Econometrics I: What is econometrics? 4/23

5 Model One of the basic paradigms in econometrics is that observations are generated by nature. Nature uses an unknown model to generate observations through a period that goes from to +. These observations form the so called population. However, we only observe a small portion of it, the sample. Based on the sample we will try to extract results that apply to the population. That means that, given our sample, we have to choose a model. A model that fits to our sample in the best possible way. Therefore, our frame is the model. And everything we will do will rely on this frame. Yves Dominicy Graduate Econometrics I: What is econometrics? 5/23

6 Model Models may be seen as economic or statistical : An economic model : It is a theoretical construction that represents economic processes (e.g. Solow model). A statistical model : It is the one that better fits the sample. The best approximation of the data generating process. Of course, they are not independent. For instance, structural models -economic- and reduced form -statistical- models in macro-econometrics are two sides of the same coin (e.g. supply and demand equations). Models can be classified in many different ways. Yves Dominicy Graduate Econometrics I: What is econometrics? 6/23

7 Model Marginal models in terms of the support of the variable of interest : Continuous real support : Gaussian, Student-t, α-stable Continuous positive support : Exponential, Gamma, Weibull, Burr The natural numbers : Poisson, Negative Binomial Discrete and finite support : Bernoulli, Binomial Conditional models in terms of the support of the variable of interest : Continuous real support : Linear regression model Continuous positive support : Non-linear models (e.g. Exponential) The natural numbers : Count data type of models (e.g. Poisson) Discrete and finite support : Discrete choice models (e.g. Probit, Logit) Yves Dominicy Graduate Econometrics I: What is econometrics? 7/23

8 Model In terms of the parameters : Parametric Semi-parametric Non-parametric In terms of the data : Cross-Section Time Series Panel Data In terms of the information : Classical Bayesian In this course, the focus will be on classical and parametric models. Yves Dominicy Graduate Econometrics I: What is econometrics? 8/23

9 Information We want to make a good use of the information that we have. Do you remember the Fisher Information? What do we have? The sample. What would we dream about? The population. Given the sample we have first to chose a model and then estimate it (e.g. parameters). Information has to do with the optimal use of the sample to get the best possible estimates. In other words, the Fisher Information is about measuring the amount of information carried by the sample about an unknown parameter. This leads the way to the question : What does best possible estimates mean? Yves Dominicy Graduate Econometrics I: What is econometrics? 9/23

10 Information It, obviously, means that the model is the best one, that the estimates are also the best ones. 1 That the model is the best one means that it fits the sample, i.e. it is able to mimic the sample. And if the sample is representative of the population, the model can also mimic the population. 2 That the estimates are the best means in statistical terms that they are consistent and that they are efficient. Consistency means that as we increase our sample, the point estimates approach to the true values. Efficiency means that the fluctuation of the estimate if we change the sample is the smallest. Hence, given a model, the best estimate will be the one that makes the best use of the information. Yves Dominicy Graduate Econometrics I: What is econometrics? 10/23

11 Risk Do you remember least squares? In least squares we minimized something. This something is the risk. Do you remember maximum likelihood? In maximum likelihood we minimized something. This something is also the risk. The risk involved in estimating a particular parameter is a measure of the degree to which the estimate is likely to be inaccurate. Suppose the estimates are consistent, what else do we want? Minimum variance! And what does minimum variance mean? Risk and Information are very related. The more information the smaller the variance. Yves Dominicy Graduate Econometrics I: What is econometrics? 11/23

12 Outline 1 2 Yves Dominicy Graduate Econometrics I: What is econometrics? 12/23

13 Econometrics What one can hear and read about econometrics : The aim in econometrics is to give an empirical context to theory. Broadly speaking, it is the application of statistical techniques to problems in economics. An econometrician can be seen as a statistician who develops the statistical techniques for solving empirical issues related to economic theory. Yves Dominicy Graduate Econometrics I: What is econometrics? 13/23

14 Econometrics What does econometrics mean? Ragnar Frisch, Editor s Note, Econometrica Vol. 1, No. 1 (Jan. 1933), 1-4. Experience has shown that each of the three view-points, that of statistics, economic theory, and mathematics, is a necessary, but not by itself a sufficient, condition for a real understanding of the quantitative relations in modern economic life. It is the unification of all three that is powerful. And it is this unification that constitutes econometrics. It is the interaction of these three that makes econometrics interesting, challenging and perhaps difficult. Yves Dominicy Graduate Econometrics I: What is econometrics? 14/23

15 Econometrics So this course is not really about econometrics...but about statistical economics. Note that within economics, econometrics has often been used for statistical methods in economics, rather than mathematical economics. Or differential calculus, optimization, game theory, to name a few, are mathematical tools to represent theory and analyse problems in economics. However, there are several interesting examples in which statistical economics, economic theory and mathematical economics dovetail nicely. Yves Dominicy Graduate Econometrics I: What is econometrics? 15/23

16 Example 1 : Random Utility Models Given a choice between 2 alternatives (j = 1, 2), let the utility that the ith person derives from, say, alternative 1 be represented by U 1. Suppose that this utility is a linear function of H number of factors. R factors (X ir ) are specific to the individual and have nothing to do with the nature of the choice and S are only choice specific (W js ). The utility function can be written as U ij = βx i + γw j + ε ij = Z ij + ε ij This is a random utility model where there is a deterministic component of utility and a random component of utility. Yves Dominicy Graduate Econometrics I: What is econometrics? 16/23

17 Example 1 : Random Utility Models The alternative with the higher utility is chosen. We observe y i = 1 if U i1 > U i0. Its probability is where F is the cdf of ε i0 ε i1. P(y i = 1) = P(U i1 > U i0 ) = P(Z i1 + ε i1 > Z i0 + ε i0 ) = P(ε i0 ε i1 < Z i1 Z i0 ) = F(Z i1 Z i0 ) Different specifications for the distribution of the error term give different c.d.f. and hence different discrete choice models. Yves Dominicy Graduate Econometrics I: What is econometrics? 17/23

18 Example 2 : Growth and panel data Solow growth model : Y (t) = K (t) α (L(t)A(t)) 1 α. Output is a function of capital, labour and technology. This model can be written in its steady state form : ln Y (t) L(t) = ln A(0) + gt + α 1 α ln s α ln(n + g + ρ) 1 α where n and g are the growth rates of labour and technology, and ρ is the capital depreciation rate and A(0) is the initial level of technology. Yves Dominicy Graduate Econometrics I: What is econometrics? 18/23

19 Example 2 : Growth and panel data The term ln A(0) + gt has two components. Since the exogenous rate of technological process, g, is thought to be the same for all countries, the term gt varies with time but not with countries. However, A(0) reflects not just technology but resource endowments, climate, institutions and so on. It may therefore differ across countries. It captures country heterogeneity. We therefore need an econometric model to estimate the economic model above that includes a term that varies with countries and a term that varies with time. The appropriate model is a dynamic panel data model with fixed time and country effect. Yves Dominicy Graduate Econometrics I: What is econometrics? 19/23

20 Example 3 : Adverse selection in stock markets Market microstructure models typically distinguish between informed and uninformed traders. There is adverse selection if a uninformed trader trades against an informed trader. Since trader identity is never revealed, the information must be content in trades. Market participants learn from the sequence of trades, update their beliefs, and this causes prices to move. Yves Dominicy Graduate Econometrics I: What is econometrics? 20/23

21 Example 3 : Adverse selection in stock markets The market cannot directly differentiate between informed and uninformed traders. However, it is usually assumed that different aspects of the trade may influence on the estimation of the probability of it being information motivated : The time of its occurrence The trade size (quantity) The time elapsed since the previous trade (frequency) The type of order submitted (aggressiveness) Liquidity providers arrive randomly according to a Poisson distribution. Informed traders enter the market only after observing a private signal. Yves Dominicy Graduate Econometrics I: What is econometrics? 21/23

22 Example 3 : Adverse selection in stock markets We have therefore to model the price arrival times, or price durations, as a function of trade information. A model for trade durations is where d i = t i t i 1, E(ε i ) = 1. d i = exp(ψ i + βz i 1 )ε i ψ i = ω + α 1 d i 1 + α 2 ψ i 1 z i 1 is a set of variables that explain adverse selection. For instance Number of transaction per second, Spread. Yves Dominicy Graduate Econometrics I: What is econometrics? 22/23

23 Summary All these models are parametric. The first one is cross-section, the second is panel and the last one is time series. But all have parameters, and distributional assumptions, that need to be estimated. And estimation means uncertainty and hence risk. Risk that has to be minimized choosing appropriately the estimated parameters. Everything in this course goes around this idea. Yves Dominicy Graduate Econometrics I: What is econometrics? 23/23