It is difficult to make predictions, especially about the future. Niels Henrik David Bohr (1885 – 1962) Danish physicist, a pioneer of the quantum theory Michal Houda Introduction to Econometrics
Introduction to Econometrics Selected Topics Michal Houda University of South Bohemia in České Budějovice Department of Applied Mathematics and Informatics Econometric Seminar, České Budějovice April 4, 2014 Michal Houda Introduction to Econometrics
Econometrics Econometrics What Is Econometrics? Michal Houda Introduction to Econometrics
What Is Econometrics economic theory qualitative analysis economic relationships microeconomics, macroeconomics Example: how the wage, inflation rate, unemployment, production, etc. are related Michal Houda Introduction to Econometrics
What Is Econometrics Michal Houda Introduction to Econometrics
What Is Econometrics Economic Statistics measuring, numerical expression of economic quantities quantitative analysis Example: the price of fuel has augmented by 1,50 € last month Michal Houda Introduction to Econometrics
What Is Econometrics Michal Houda Introduction to Econometrics
What Is Econometrics Mathematical Economics mathematical models for example, (systems of) mathematical equations Michal Houda Introduction to Econometrics
What Is Econometrics Example Economic theory: household expenditures on a certain product of long-term consumption depends (somehow) on the total household expenditures, and on the price of the product Mathematical Economics: formulates the hypothesis by a mathematical model LE = a · TE + b · P where LE . . . long-term household expenditures on the selected product TC . . . total household expenditures P . . . price of the product a , b . . . unknown parameters Michal Houda Introduction to Econometrics
What Is Econometrics Michal Houda Introduction to Econometrics
What Is Econometrics Mathematical Statistics mathematical modelling of different relationships between variables “mathematically” defines the measurement and investigates the methodology of this measurement Michal Houda Introduction to Econometrics
What Is Econometrics Example LE = a · TE + b · P Let t = 1 , . . . , 10 years . . . time, time intervals ε t . . . error, disturbance, residuum, “remainings” in t th year Statistical (econometric) model of the problem: LE t = a · TE t + b · P t + ε t ∀ t Mathematical statistics provides methods to discover (estimate) the unknown parameters a and b . Michal Houda Introduction to Econometrics
What Is Econometrics Michal Houda Introduction to Econometrics
What Is Econometrics Econometrics linking economics, mathematics, and statistics discipline that using mathematical models, statistical methods, and empirical data (given by measurements or observations) verifies the conclusions of the economic theory or try to discover such relationships Michal Houda Introduction to Econometrics
Econometric model Econometric model LE t = a · TE t + b · P t + ε t ∀ t created by a mathematical (statistical) formulation of a certain economic hypotheses if we have data : year ( t ) 1 2 3 4 5 6 7 8 9 10 long-term exp. ( LEt ) 3 4 5 6 7 8 9 10 12 14 total exp. ( TEt ) 15 20 30 42 50 54 65 72 85 90 price ( Pt ) 10 10 10 7 7 7 6 5 6 4 we can, by different (statistical, econometric) methods , for example, by the least squares method (OLS): ε 2 min → � t estimate parameters a and b of the model, and detect some other information about the quality of this estimate (on the computer: a = 0 , 14 and b = 0 , 08, and we have also found that b is not statistically significant, that is, the dependency of long-term expenditures LE on the price P is negligible) Michal Houda Introduction to Econometrics
Differences from Related Disciplines the data are not experimental, but observational (retrospective), usually 1 random the model assumptions set by the “nature” (not by the researcher), or they 2 are not always fulfilled the simplifications commonly used (and a correct interpretation of results is 3 necessary). Five Steps in Empirical Analysis qualitative analysis of the problem ⇒ economic model 1 mathematical/statistical transformation ⇒ econometric model 2 data ⇒ quantitative ( empirical ) analysis of the problem 3 economical/statistical verification of the model ⇒ testing the hypothesis 4 implementation/further exploration of the model 5 Michal Houda Introduction to Econometrics
The Structure of Economic Data cross-sectional data 1 a set of objects (individuals, households, firms, cities, . . . ) taken at a given point in time usually obtained by random sampling ordering of the data is not important time series data 2 observation of variables over time chronological ordering potentially important information issues: (serial) dependence, seasonal patterns, . . . pooled cross sections 3 cross-sectional data taken over time new sample taken at each time panel (longitudinal) data 4 dtto, but time series for each cross-sectional object Michal Houda Introduction to Econometrics
The Question of Causality economist’s goal: establish a causal effect between variables, under ceteris paribus = “other (relevant) factors being equal” not easy task number of affecting factors may be immense some of them cannot be held “fixed” in practice (e. g. education, experience) have we chosen enough factors to be held “fixed” Econometrics: how to simulate a ceteris paribus experiment to estimate the causality of variables in question Michal Houda Introduction to Econometrics
Definition of the Simple Linear Regression (SLR) Model “explaining y in terms of of x ” y = β 0 + β 1 x + ε (1) y x ε dependent variable independent variable error term explained variable explanatory variable disturbance response variable control variable unobserved (other) factors predicted variable predictor variable residual term regressand regressor (covariate) Convention Y , X . . . random variable representing regressand, regressors (in econometrics often also written as y , x ) y , x . . . observed (e. g. measured) values of Y , X ˆ . . . calculated or predicted values of regressand y . . . (non-observed) error or disturbance ε ˆ . . . calculated residuals ε Michal Houda Introduction to Econometrics
Definition of the Simple Linear Regression Model Linearity and Issues Linearity: the effect of x on y is linear – if ∆ ǫ = 0 then β 1 = ∆ y ∆ y = β 1 ∆ x , i. e. ∆ x β 1 . . . slope parameter β 0 . . . intercept parameter Issues the relationship never exact 1 the functional relationship may not be linear 2 how we can be sure to capture a ceteris paribus relationship between y and x 3 β 1 measures the effect of x on y , other factors ε held fixed but, in fact, we ignore all these factors ⇒ we only get (more or less) reliable estimators of β 0 , β 1 , based on random sample under some (restrictive) assumption. Michal Houda Introduction to Econometrics
Deriving the Ordinary Least Squares Estimates Least Squares Method Under zero conditional mean assumption (see later): E [ y | x ] = β 0 + β 1 x . . . population regression function (PRF) Least Squares Method : n n β 1 x t ) � 2 � ε 2 � y t − (ˆ β 0 + ˆ � min → SSR := ˆ t = t =1 t =1 By deriving & setting the derivatives = 0, we obtain � ( x t − ¯ x )( y t − ¯ y ) β 1 = sample cov( x , y ) ˆ ˆ y − ˆ = , β 0 = ¯ β 1 ¯ x � x ) 2 sample var x ( x t − ¯ y = ˆ β 0 + ˆ ˆ β 1 x (2) . . . OLS regression line / sample regression function (SRF) Michal Houda Introduction to Econometrics
Statistical Properties of the OLS Estimators Assumptions for Simple Linear Regression Assumption SLR 1 (Linearity in parameters) In the population model, y is related to x and ε as y = β 0 + β 1 x + ε (3) Assumption SLR 2 (Random Sampling) We have the random sample ( x 1 ; y 1 ) , . . . , ( x n ; y n ) from the population model given by (7). Assumption SLR 3 (Sample Variation in the Explanatory Variable) The sample outcomes on x (namely { x t , t = 1 , . . . , n }) are not of the same value. Assumption SLR 4 (Zero Conditional Mean) E [ ε | x ] = 0 Michal Houda Introduction to Econometrics
Statistical Properties of the OLS Estimators Unbiasedness of OLS Theorem 1 (Unbiasedness of OLS) Under SLR.1–4, E ˆ β 0 = β 0 and E ˆ β 1 = β 1 for any values of β 0 , β 1 . In other words, β 0 is unbiased estimator of β 0 and ˆ ˆ β 1 is unbiased estimator of β 1 . Remarks Unbiasedness is the property of sampling , not of the sample 1 if the sample is typical, we obtain good estimators, but not always, we are so “lucky”, and we never know if we are lucky Unbiasedness fails if any of SLR’s assumptions fails 2 the model is not (technically) linear (SLR.1) the data are time series with serial correlation , or the sample is not representative (SLR.2) if SLR.3 fails we cannot even obtain OLS estimators if SLR.4 fails the OLS estimators will probably be biased Michal Houda Introduction to Econometrics
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