Applied Econometrics - QEM Theme 1: Introduction to Econometrics Chapter 1 + Probability Primer + Appendix B in PoE

Transcription

1 Applied Econometrics - QEM Theme 1: Introduction to Econometrics Chapter 1 + Probability Primer + Appendix B in PoE Warsaw School of Economics

2 Outline 1. Introduction to econometrics 2.

3 Denition of econometrics Econometrics: Application of mathematical and statistical techniques to economics in the study of problems, the analysis of data, and the development and testing of theories and models.

4 The role of econometrics in economics As economists we think in terms of theoretical concepts such as scarcity, opportunity cost, price elasticity, etc. As econometricians we evaluate how economic concepts are supported by empirical data Econometrics lls the gap between being a theoretical and applied economist As econometricians we should be able to say to our employers: I can predict the sales of your product I can estimate the eect on your sales if your competition lowers its price by 1$ per unit I can test whether your new ad campaign is actually increasing your sales

5 Some examples of research questions 1 A city council ponders the question of how much violent crime will be reduced if an additional million dollars is spent putting uniformed police on the street. 2 The owner of a local Pizza Hut must decide how much advertising space to purchase in the local newspaper, and thus must estimate the relationship between advertising and sales. 3 Louisiana State University must estimate how much enrollment will fall if tuition is raised by $300 per semester, and thus whether its revenue from tuition will rise or fall. 4 The CEO of Proctor & Gamble must estimate how much demand there will be in ten years for the detergent Tide, and how much to invest in new plant and equipment.

6 Stages of building an econometric model 1 Setting up a research hypothesis (including literature review) 2 Choosing a functional form and the set of explanatory variables 3 Collecting the data 4 Estimating the model 5 Verication process 6 Application Example: estimation of the Phillips curve for Poland Research hypothesis: To estimate the slope of the Phillips curve in Poland Model specication: π t = α 1 + α 2u t + ɛ t Data: Quarterly data for ination and the unemployment rate from the period (source: CSO). Estimation: Ordinary Least Squares (OLS) Verication: Determination coecient R 2, test whether α 2 < 0 Application: Verication of the theoretical model

7 Specication of econometric model The general specication of a (single-equation) econometric model: y t = f (x t, α, ɛ t ) for t = 1, 2,..., T where: y t - a dependent variable (e.g. sales of a product); x t - a vector of K independent variables (e.g. price, price of substitutes / complements, income); α - a vector of model parameters (unknowns that are to be estimated); ɛ t - error term (stochastic part of the model); t - moment of observation For linear models the notation translates into: y t = α 1 x 1t + α 2 x 2t α K x Kt + ɛ t

8 Econometric versus economic model 1 Variables in the econometric model are indexed by t. 2 There is a stochastic part represented by the error term in the econometric model. Example: a model for the marginal propensity to consume (MPC) out of disposable income. Economic model: C = α + βy Econometric model: C t = α + βy t + ɛ t

9 Types of data 1 Classication based on the source of the data: macroeconomic data (macroeconometrics) microeconomic data (microeconometrics) nancial data (nancial econometrics) experimental data (experimental econometrics) 2 Classication based on the type of data sample: time series (collected over discrete intervals of time: y t, t = 1, 2,..., T ) cross-section data (collected across sample units in a particular time period: y i, i = 1, 2,..., N) panel or longitudinal data (observations on many individual units over time: y it, i = 1, 2,..., N and t = 1, 2,..., T )

10 Outline 1. Introduction to econometrics 2.

11 Random variable Random variable A variable whose value is unknown until it is observed Discrete var.: Indicator var.: Continuous var.: takes only a limited, or countable, number of values takes the values 1 if yes, or 0 if no take any value in an interval

12 Probability distribution 1 Probability density function (pdf): Discrete var.: f (x) = P(X = x) and f (x i ) = 1 Continuous var.: since P(X = x) = 0 the pdf can be interpreted as the relative probability, so that P(a X b) = b f (x)dx and a f (x)dx 2 Cumulative density function (cdf): F (x) = P(X x)

13 Joint, marginal and conditional pdf (for discrete vars.) 1 Joint pdf 2 Marginal pdf f (x, y) = P(X = x, Y = y) f X (x) = P(X = x) = y f (x, y) 3 Conditional pdf 4 Statistical independence f (y x) = f (y x) = P(Y = y X = x) = f (x, y) f X (x) f (x, y) f X (x) = f Y (y) f (x, y) = f X (x)f Y (y)

14 Expected value Expected value of a variable is given by its mathematical expectation Discrete var.: µ = E(X ) = x i f (x i ) Continuous var.: µ = E(X ) = xf (x)dx Important: Expected value (µ X ) is called population mean sample average ( x) Conditional EV Discrete var.: Continuous var.: Two important rules for EV: µ X Y = E(X Y = y) = x i f (x i y) µ X Y = E(X Y = y) = xf (x y)dx E[g(X )] = g(x)f (x) E(aX + b) =ae(x ) + b

15 Variance and covariance The variance of a rand. var. is the expected value of: var(x ) = σx 2 = E[(X µ) 2 ] = f (x)(x µ) 2 dx The covariance between X and Y is the expected value of: cov(x, Y ) = σ XY = E[(X µ X )(Y µ Y )] = xy f (x, y)dxdy The correlation between X and Y is: cor(x, Y ) = ρ XY = σ XY σ X σ Y 2 important rules: var(ax + b) =a 2 var(x ) var(x + Y ) =var(x ) + var(y ) + 2cov(X, Y )

16 Important probability distributions: Normal distribution For X N(µ, σ 2 ) the pdf is: ( ) 1 (x µ) 2 f (x) = 2πσ 2 exp σ 2 Standardization to Z N(0, 1): Calculating probabilities: ( a µ P(a X b) = P σ where Φ is cdf for Z Z = X µ σ N(0, 1) Z b µ ) = Φ σ ( ) ( ) b µ a µ Φ σ σ

17 Important probability distributions: Chi-square, t and F distributions For independent Z i N(µ, σ 2 ) the variable: V = Z1 2 + Z Zm 2 χ 2 (m) has a χ 2 distribution with m degrees of freedom, where: E(V ) = m and var(v ) = 2m For independent Z N(0, 1) and V χ 2 (m) the variable: t = Z V /m t (m) has the t-distribution with m degrees of freedom. A method to generate numbers from t-distribution: (step i) draw v from χ 2 (m), step (ii) draw t from N(0, v/m) For large m the χ 2 (m) and t (m) converge to a normal distribution.

18 Important probability distributions: Chi-square, t and F distributions For independent V 1 χ 2 (m 1 ) and V 2 χ 2 (m 2 ) the variable: F = V 1/m 1 V 2 /m 2 F ( m 1, m 2 ) has the F -distribution with m 1 and m 2 degrees of freedom. Note that if Y t (m) then Y 2 F (1, m)