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Table of Contents CHAPTER 1 – INTRODUCTION................................................................................................................................... TYPES OF DATA....................................................................................................................

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Table of Contents CHAPTER 1 – INTRODUCTION................................................................................................................................... 3 TYPES OF DATA..............................................................................................................................................................3 STEPS IN ECONOMETRIC ANALYSIS...............................................................................................................................3 DATA STRUCTURES........................................................................................................................................................3 CHAPTER 2 – SIMPLE REGRESSION MODEL............................................................................................................... 5 ESTIMATION..................................................................................................................................................................6 GOODNESS OF FIT.........................................................................................................................................................9 THE GAUSSMARKOV ASSUMPTIONS FOR SIMPLE REGRESSION..................................................................................10 SIMPLE REGRESSION PROPERTIES..............................................................................................................................10 DRAWBACKS OF THE SIMPLE REGRESSION MODEL....................................................................................................12 CHAPTER 3 – MULTIPLE REGRESSION MODEL.......................................................................................................... 13 ESTIMATION................................................................................................................................................................14 GOODNESS OF FIT.......................................................................................................................................................16 THE GAUSS-MARKOV ASSUMPTIONS FOR MULTIPLE REGRESSION............................................................................17 MULTIPLE REGRESSION PROPERTIES..........................................................................................................................18 OMITTED VARIABLES..................................................................................................................................................19 IRRELEVANT VARIABLES..............................................................................................................................................20 CHAPTER 4 – MULTIPLE REGRESSION ANALYSIS - INFERENCE................................................................................... 21 LAST ASSUMPTION FOR MULTIPLE REGRESSION........................................................................................................21 IMPLICATIONS OF THE NORMALITY ASSUMPTION.....................................................................................................21 THE SAMPLING DISTRIBUTION....................................................................................................................................22 HYPOTHESIS TESTING – SINGLE PARAMETER..............................................................................................................22 STATISTICAL VS ECONOMICAL SIGNIFICANCE.............................................................................................................25 CONFIDENCE INTERVALS.............................................................................................................................................26 HYPOTHESIS TESTING – SEVERAL PARAMETERS..........................................................................................................26 TESTING MULTIPLE HYPOTHESISES.............................................................................................................................27 CHAPTER 5 – MULTIPLE REGRESSION ANALYSIS: OLS ASYMPTOTICS........................................................................29 CONSISTENCY.............................................................................................................................................................29 NORMALITY ASSUMPTION UNDER ASYMPTOTICS......................................................................................................31 LARGE SAMPLE INFERENCE.........................................................................................................................................32 EFFICIENCY................................................................................................................................................................. 33 CHAPTER 6 – FURTHER ISSUES ON MULTIPLE REGRESSION...................................................................................... 34 DATA SCALING.............................................................................................................................................................34 FUNCTIONAL FORMS..................................................................................................................................................35 GOODNESS OF FIT..........................................................................................................................................................36 PREDICTION................................................................................................................................................................36 CHAPTER 7 – QUALITATIVE INFORMATION WITH MULTIPLE REGRESSION................................................................38 SINGLE INDEPENDENT DUMMY VARIABLE.................................................................................................................38 MULTIPLE DUMMY VARIABLES...................................................................................................................................38 THE LINEAR PROBABILITY MODEL...............................................................................................................................39 CHAPTER 8 – HETEROSCEDASTICITY........................................................................................................................ 40 HETEROSCEDASTICITY ROBUST INFERENCE AFTER OLS..............................................................................................40 TESTING FOR HETEROSCEDASTICITY...........................................................................................................................40

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WEIGHTED LEAST SQUARES METHOD...........................................................................................................................41 CHAPTER 9 – MORE ON SPECIFICATION AND DATA ISSUES...................................................................................... 43 FUNCTIONAL FORM MIS-SPECIFICATION....................................................................................................................43 NONNESTED MODELS.................................................................................................................................................43 PROXY VARIABLES FOR UNOBSERVED EXPLANATORY VARIABLES...............................................................................44 PROPERTIES OF OLS UNDER MEASUREMENT ERROR.................................................................................................45 MISSING DATA............................................................................................................................................................46 NON-RANDOM SAMPLING.........................................................................................................................................46 OUTLIERS....................................................................................................................................................................46 CHAPTER 10 – BASIC REGRESSION ANALYSIS WITH TIME SERIES DATA.....................................................................47 STATIC MODELS...........................................................................................................................................................47 FINITE DISTRIBUTED LAG MODELS..............................................................................................................................47 FINITE SAMPLE PROPERTIES.......................................................................................................................................48 The Gauss-Markov assumptions for time series data..............................................................................................48 Variance of OLS.........................................................................................................................................................48 Inference for time series...........................................................................................................................................49 FUNCTIONAL FORM, DUMMY VARIABLES AND INDEX NUMBERS..............................................................................49 TRENDS AND SEASONALITY........................................................................................................................................49 CHAPTER 11 – FURTHER ISSUES IN USING OLS WITH TIME SERIES DATA...................................................................51 STATIONARY AND WEAKLY DEPENDENT TIME SERIES.................................................................................................51 ASYMPTOTIC PROPERTIES OF OLS..............................................................................................................................52 HIGHLY PERSISTENT TIME SERIES................................................................................................................................53 CHAPTER 13 – POOLING CROSS SECTIONS ACROSS TIME: SIMPLE PANEL DATA METHODS........................................54 POOLING INDEPENDENT CROSS SECTIONS ACROSS TIME..........................................................................................54 PANEL DATA................................................................................................................................................................54 CHAPTER 14 – ADVANCED PANEL DATA METHODS.................................................................................................. 56 FIXED EFFECTS ESTIMATOR.........................................................................................................................................56 RANDOM EFFECTS MODEL.........................................................................................................................................57 FORMULAS............................................................................................................................................................ 58 CHAPTER 10: TIME SERIES..........................................................................................................................................58 CHAPTER 13: POOLING CROSS SECTIONS ACROSS TIME: SIMPLE PANEL DATA METHODS..........................................59

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CHAPTER 1 – INTRODUCTION Econometrics use statistical methods to analyse data. Its goal is to estimate relationships between economic variables: - Testing economic theories and hypotheses - Forecasting economic variables - Evaluating and implementing government and business policy

TYPES OF DATA 1. Observational data: we observe what individuals/firms/municipalities/etc. do. 2. Experimental data: we control what individuals/firms/municipalities/etc. are exposed to.

STEPS IN ECONOMETRIC ANALYSIS 1. A question – often based on an economic model (theory) 2. A method – an econometric model (to be estimated) 3. A hypothesis – specification of (null) hypothesis that we want to test

DATA STRUCTURES 1. Cross-sectional data o A sample of units of observations at a given point in time o Observations are more or less independent (random sampling from an underlying population) o Typically encountered in applied microeconomics

2. Time series data o Observations of a variable or several variables over time, i.e. one unit of observation in several time periods o Typically encountered in applied macroeconomics and finance o Time series observations are typically serially correlated. o Typical features of time series include trends and seasonality.

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3. Pooled cross-sectional data o Two or more cross sections are combined in one data set o Cross sections are drawn independently of each other so there is no link between units of observations across samples o Often used to evaluate reforms

4. Panel/Longitudinal data o The same cross-sectional units are followed over time. Allows us to compare a unit to himself/herself at two different points in time with varying policy exposure o Panel data have a cross-sectional and a time series dimension o Panel data can be used to account for time-invariant unobservables

Often, we are interested in uncovering the casual impact of one variable (x) on another variable (y). The effect of x on y when the other factors are held constant is called to be the ceteris paribus (=other things being equal). If there is correlation between the variables, that does not necessarily mean that there is also causality. Causality can be claimed if the researcher gets to control who are exposed to which values of x. An experiment is a way to control x, the golden standard in effect evaluation is the usage of a Randomized Controlled Trial (RCT). It is attempted to build an empirical analysis that replicates what the experiment does. This forces one to be explicit about the different scenarios to be compared (the Q). This forces one to think about what variables we need to control for in order to make a good case for causality. Any econometric model must have behind it some THEORY of causation.

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CHAPTER 2 – SIMPLE REGRESSION MODEL Simple/two variable/bivariate linear regression model tries to explain y in terms of x. y=β 0 + β 1 x +u y : dependent variable, explained variable, response variable, predicted variable, regeressand β 0 : intercept (the value of y for x=0) β 1 : slope parameter, describing the relationship between y and x, holding the other factors fixed: ∆ y= β1 ∆ x . Thus, the change in y is simply β 1 multiplied by the change in x. x : independent variable, explanatory variable, control variable, predictor variable, regressor u : error term, disturbance, unobservables Systematic part of y: Unsystematic part of y:

β 0+ β 1 x u

The simple regression model does not allow us to draw casual conclusions about how x affects y. Restrictions need to be imposed on how the error term is related to the explanatory variable. - As long as the intercept ( β 0 ) is included in the equation the following assumption can be made: E ( u) =0 That is the average of u in the population is 0. - It is not enough to assume that u and x are uncorrelated, Correlation measures the strength of a linear relationship between two variables. So assuming only uncorrelation, results in disregarding other functional forms, such as the quadratic form (x 2). Pearson’s correlation coefficient: E[ ( x−μ x )( u−μu ) ] r xu= σx σu Instead it is assumed that the average value of u does not depend on the value of x: E ( u|x )=E ( u ) i.e. u is mean independent of x. - The zero conditional mean assumption is the combination of the previous two assumptions: E ( u|x )=E ( u ) =Cov (u , x ) =0 The explanatory variable must not contain information about the mean of the unobserved factors for the researcher to claim causality. It is possible for u to be uncorrelated with x while being correlated with functions of x, such as x2. E.g.

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The conditional mean independence implies that: E ( y|x )=E ( β 0 + β 1 x +u |x ) =β 0 + β 1 x + E ( u|x )= β 0 +β 1 x +0 = β 0 + β 1 x This is called the population regression function (PRF), which is linear in x. This means that the average value of the independent variable can be expressed as a linear function of the explanatory variable. The interpretation of β 1 is that a one-unit change in x changes the expected value of y by the amount of β 1 . The PRF is unknown but can be estimated using a random sample of the population.

ESTIMATION Two methods 1. Use the zero conditional mean assumption to define a method of moments estimator of the regression function 2. Derive the estimates by minimizing the sum of the squared regression residuals. This method is called the least squares method To estimate the regression model one needs data, a random sample of n observations: x ¿ (¿ i, y i ¿ ):i=1,2 , … , n } . The population regression equation y=β 0 + β 1 x +u can be written ¿ ¿ y i=β 0 + β 1 x i +ui for each I as: Method of moments Using the linearity property of expectations E ( X +Y )= E ( X ) + E(Y ) and the fact that the average of u in the population is 0 E ( u)=0 we get the covariance (2nd moment) between x and u is 0: Cov (x ,u ) =E [ [ x−E ( x ) ][u −E ( u ) ] ]=E [ xu − xE ( u )−E ( x ) u+ E ( x )E ( u ) ] =E ( xu )−E ( x ) E ( u )−E ( x ) E ( u) +E The two equations E ( u) =0 and Cov (x ,u ) =E ( xu )=0 can be rewritten in terms of the observable variables x and y and the unknown parameters β 0 and β 1 in the following way: E ( y i−β 0− β 1 xi ) =0 E(x i ( y i−β 0−β 1 x i ))=0 The sample counterparts of the equations: n

−1

n

β 1 x i) =0 β 0− ^ ∑( y i −^ i=1

x ^ (¿¿ i ( y i− β 0− ^ β 1 xi ))=0 n

−1

n

∑¿ i=1

Solve for n

^ β 0 from the first equation:

n−1 ∑( y i −^ β 0− ^ β 1 x i) =0 i=1

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´y − ^ β1 x´ =0 β 0− ^ n

Where

´y =n−1 ∑ ( y i ) i=1

´y − ^ β 1 x´ = ^ β0 Use this to solve the other equation: n

−1

n

−1

n

−1 and ´x =n ∑ ( x i) i=1

n

∑( x i( y i−( ´y − ^β 1 ´x)− ^β1 x i)) =0 i=1 n

∑( x i ( y i− ´y + ^β1 ´x −^β 1 x i ))=0 i=1

Since n−1 does not affect the solution, it can be dropped. n

n

(xi ( y i− ´y ) ) =β^1 ∑ ( x i( x i− x´ ) ) ∑ i=1 i=1 xi −´x n

n

( ¿ ( y i− ´y ) )= ^ β1 ∑ ( x i−x´ ) 2 ∑ i=1 i=1

See A7 and A8 in math refresher for more details on the summation operator x i−´x n

^ β 1=

∑ ( ¿( y i − ´y ) ) i=1 n

∑ ( x i−´x )2 i=1

Thus, the MM estimates of ^ β 0= ´y − ^ β1 x´ x i−´x

^ β 0 and ^ β1

are given by:

n

^ β 1=

∑ ( ¿( y i − ´y ) ) i=1 n

2 ∑ ( x i−´x ) i=1

=

Cov (xi , yi) Var (x i)

If x i and y i are positively/negatively correlated in the sample, then ^ β 1 is positive/negative. Ordinary least squares method This method involves minimizing the sum of the squared regression residuals.

Regression residual: the difference between the observed value and the estimated value:

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^ β 1 xi ui = y i − ^ y i= y i− ^ β 0− ^ Minimize the sum of the squared egression residuals: y i− ^ β 0− ^ β 1 xi ¿ ¿ ¿ n

n

i=1

i=1

2 min^β , ^β =∑ ^ ui =∑ ¿ 0

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First order conditions (FOC’s): n

∂ =−2 ∑ ( yi − ^ β 0−^ β 1 x i) =0 β0 ∂^ i=1 n ∂ =−2 ∑ x i ( yi − ^ β 0− ^ β 1 x i) =0 ∂^ β1 i=1 β 0 first we can get rid of the -2 by dividing both sides with -2: To solve for ^ n

∂ =−2 ∑ ( yi − ^ β 0−^ β 1 x i) =0 β0 ∂^ i=1 n

β 1 x i ) =0 β 0− ^ ∑ ( y i−^ i=1 n

∑ i=1

n

n

i=1

i=1

β 0 −∑ ^ y i−∑ ^ β 1 x i=0

β 0−n∗ ^ n∗ ´y −n∗^ β 1∗ ´x =0 n∗( ´y − ^ β...