Page 1 Page Regression model Theme 2 Michał Rubaszek Michał Regression Model Regression Chapters from 2to 6 of PoE Applied Econometric QEM Based on presentation Paczkowski R. Walter by

Applied Applied Econometrics QEM Page 2 Page Regression model Economic andEconomic Econometric Model

Applied Applied Econometrics QEM ’’ or ’’ Page 3 Page independent depends on depends y , which, us give the ‘‘ the 2 x σ and we need to use we data to andneed ’’ and ’’ Regression model random variable is a is y dependent variable ’’ variable ’’ and the conditional variance conditionalthe and the ‘‘ the y|x µ y x )= x | y ( explanatory valuable information about the population we we considering are population the about information valuable to learn about the relationship theabout learnto mean conditional calculate to helps model econometric The E In econometrics ‘‘ income income call We Economists interested in relationships between variables between in relationships interested Economists Example: theexpendituresus that theorytells ‹ ‹ ‹ ‹ ‹ Applied Econometrics QEM y Page 4 Page = $2000 = x = $1000 and = x Regression model givenincomes Figure 2.1b Probability distributions of food distributions2.1bProbabilityof Figureexpenditures

Applied Applied Econometrics QEM Page 5 Page economic value: x econometric econometric model: | Regression model x dx given angiven (|) (|) ∆ y Eyx dEyx ∆ = = 2 β and a corresponding anda – intercept – slope 1 2 β β To investigate the relationship wetherelationship investigate To buildan Interpretation–slopethe of derivativethe of of expectedvalue model

• Applied Econometrics QEM Page 6 Page Regression model averagepersonper expenditureincome and food Figure 2.2 Theeconomic model:relationshiplinear abetween 2.2Figure

Applied Applied Econometrics QEM Page 7 Page at two levelsincome attwo of y Regression model Figure 2.3 Thefor probabilitydensityfunction 2.3 Figure 2.2 Applied Applied Econometrics QEM Model An Econometric Econometric An Page 8 Page ) ( upon the expected expected uponthe k K K x K Regression model other xs held constant held xs other ) k k ( x x ∆ ∂ Ey Ey ∆ ∂ 1 22 33 ββ β β = = =+ + ++ + k , all otherconstantall held, variables (ceteris paribus) β y xx xe y measures the effect of a in change theeffect measures k value of value Multiplemodel regression – ageneral case: β Applied Econometrics-QEM Eq.5.3 ) 9 Page ∼ (0, , Regression model Economiceconometric vs. model Economic model Econometric model Applied Econometrics QEM Page 10 Page is: is: e ≠ is: , is: , e for for x j ) e 0 and i , e Regression model ∼ (0, normallydistributed: is e , , valuefor of each y 0 ↔ is notrandomistakesand 2different at least term term x A4: The covarianceThe between A4: Assumptionsoflinear econometric model: valueThe of A1: A2: The expected valueThe of randomtheerror A2: Variable A5: values A6+:Random A3: The variancetheTheerror of random A3: Applied Econometrics QEM Page 11 Page 2 2 , 1, , Regression model are not random and are not exact and are not random are tk L x 2 L LK ASSUMPTIONS of the Multiple Regression Model Regression the ASSUMPTIONS of Multiple = = 1 2 2 1 2 2 β+β ++β σ⇔ σ = = σ    =β+β + +β ⇔ = ij ij 1 2 2 i i yy ee y e i i K iK i ~ ( ), ~ (0, ) =β+β + +β + = () ()0 i i K iK i i i K iK i y N x x e N y x x e i N Ey x xEe var( ) var( ) cov( , ) cov( , ) 0 Assumptions a multiple for regression model: A1. A2. A3. A4. of each The values A5. linear functions of the other explanatory variables otherexplanatory of functionsthe linear A6. Applied Econometrics QEM Page 12 Page y and e Regression model Figure 2.4 Probability densityfunctions2.4ProbabilityforFigure

Applied Applied Econometrics QEM Page 13 Page Regression model Estimating the RegressionParameters theEstimating

Applied Applied Econometrics QEM Page 14 Page Regression model Table 2.1 Food Expenditure and Data 2.1Food ExpenditureIncome Table

Applied Applied Econometrics QEM Page 15 Page Regression model Figure 2.6DataFigurefor expenditure example food

Applied Applied Econometrics QEM ) , Page 16 Page ( i x : 2 ) b − 1 b − fitted values i y minimizeresiduals: ofsquared sum the = 1 2 i Regression model ˆ y and = + − i i i we canwe calculate y ˆ y b bx ( = i ˆ e and residuals: For anyFor values valuessquares least The of and

Fitted values,residuals andleast squares Applied Econometrics QEM Page 17 Page , ê and theline regressionandfitted,ê y Regression model Figure 2.7 The relationshipamong 2.7Figure

Applied Applied Econometrics QEM 2 andb 1 Page 18 Page imizing values b Regression model Figure 2A.1 The sum sum TheFigure 2A.1squaresof functionand the min

Applied Applied Econometrics QEM are 2 β x 2 and b Page 19 Page 1 β − ) y = 1 b and ) Regression model y ( − 2 ) i y x )( − i x x − ( i x ∑ ( ∑ = obtained my minimizing the sum sum the my minimizing obtained Least squares estimates for the unknown unknown thefor parameters squaresestimates Least Solution for one explanatoty variable case: 2 b

Least squares estimator Applied Econometrics QEM and Page 20 Page ⋯ Regression model ′ , , don'tbutknow of thevalues - the vector of explanatory variables - the vector of parameters. ]′ and … … [1 in a vectorin a form: observe We Least squares estimator – multiple regression Multiple regression needestimate to it Applied Econometrics-QEM . The , the ) Page 21 Page odel (described by ( random variable ∑ and is a is anda ∑ such thatsuch the SEE is minimum Regression model that obtainwe byapplying the general ∑ so that: , we we , can find () ∑ be the estimate be the estimate of The LS estimator generalis a formula The LS estimator properties of which dependonthe structure of the m assumptions). numbers LS estimates are formulas theto observed data. Since SSE ondepends solution is the formula for LS estimator: • Let • Fitted values: Residuals: Sum Sum of residuals:sq. Applied Econometrics-QEM Page 22 Page Regression model Table 2.1 Food Expenditure and Data 2.1Food ExpenditureIncome Table Least squares estimator - example

Applied Applied Econometrics QEM Page 23 Page 4160 . 83 = ) 2096 . 6048 . ? 10 2 = 19 b )( and 2684 7876 . 2096 . 1 . b Regression model 10 ( i 1828 18671 x − = ) 21 . y 5735 . − 2 ) i 10 y x + 283 )( − i x = x − ( 42 x . i 2 x b ∑ ( − 83 y ∑ = = = i 2 1 ˆ y b b We canWe calculate: Andreport that: What interpretation of

Least squares estimator - example Applied Econometrics QEM Page 24 Page Regression model Figure 2.9EViewsFigure Output Regression

Applied Applied Econometrics QEM Page 25 Page Regression model Figure 2.8 Theregressionfitted line 2.8Figure

Applied Applied Econometrics QEM for a 61 . Page 26 Page 287 = ) = 20. We20. = obtain: 20 x ( 21 . 10 + 42 . 83 = Regression model i x 21 . 10 + 42 . 83 that a household of weeklya with a thatincome = ˆ y predict predict $2000willspend$287.61 per foodweek on householdof $2000,with income so that We Point prediction Supposewe wanted predictthattofood expenditure Applied Econometrics QEM Page 27 Page Regression model Assessing the Least Least SquaresAssessingtheFit

Applied Applied Econometrics QEM ) ∑ Page 28 Page ̅ ̅ ∑ ̅ ̅)( ∑( ∑ Regression model ̅ ) 1 ̅ ∑ ̅)( ∑ ∑(

Notice Notice thatestimators LS not (do confuse with estimates) are random variables so wecan calculate their expected values, variances, covariances or probability distributions Given that: canWe derive: Applied Econometrics QEM 2 b bution and 1 e from from e b (estimate Page 29 Page 0 imply that:

] 0 ∑ unbiased: ̅ ̅ is 2 b ∑ Regression model ∑ ][ A2 and isnotrandom x estimator).Fordifferent the samples estimates of are different – they are of estimator the just singledraws from distrithe Important: unbiasednessImportant: estimat doesnotthatsayan sampleoneanyisto parametertruevalueclosethe ≠ This meansthe thatestimator A5 [ A5 Applied Econometrics QEM are 2 2 N − 2 i and b and Page 30 Page σ x x 1 () ∑ = 2 b and increases with 2 var( ) σ the variancethe converges0 to Regression model 2 → ∞ 2 − i i x () ∑ ∑ N x x      2 what isvariance the estimator?LS the of σ = 1 b Question: A1-A5 hold If b of and covariancevariances the then var( ) estimatesof Precision decreases with Consistent for estimators: Effectiveestimators estimators: smallestthe with variance

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Applied Applied Econometrics QEM . . not ) possible Page 32 Page estimators estimators ed estimators ed er linear and sumptions A1-A5 sumptions true, then LS is because theyhave not sayabout all not Regression model Best Linear Unbiased Estimators (BLUE Estimators Unbiased Linear Best the minimum the minimum variance. unbiased estimators - Theorem does the estimators. must be true. If ofany these assumptions are thelinear best unbiasedestimator. 3. In order forTheorem hold,tothe Gauss-Markovas 2. The LS estimators theare best withintheir class 1. The LS estimators “best” are compared when to oth Notice that: Gauss-Markov theorem LS the model, regression linear the of A1-A5 Under have the smallest variance of all linear and unbiasand linear all of variance smallest the have the are They Applied Econometrics QEM Page 33 Page Regression model Interval estimation

Applied Applied Econometrics QEM in Page 34 Page Regression model ultiplemodel regression Let us focusm a on The econometric model is: modeleconometricThe which sales revenue depends on price andonpricesalesrevenuewhichdepends expenditure:advertising Applied Econometrics-QEM Page 35 Page Regression model Table 5.1 Observations on Monthly Sales, Price, and Advertising in BigAdvertisingin MonthlyPrice,on 5.1ObservationsSales,and Table Barn Burger Andy’s

Applied Applied Econometrics QEM ales ales e of $1 of e g Page 36 Page : Regression model PRICE: ADVERT with advertising held constant, an increase in pric in increase an constant, held advertising with will lead to a fall in monthly revenue of $7,908of revenue monthlyin fall a to will lead advertisin in increase an constant, held price with s in increase anto will lead $1,000of expenditure $1,863of revenue 1.on coefficient The 2.on coefficient The Interpretations Interpretations of the results:

• Applied Econometrics-QEM Page 37 Page ∑ ) ) , Regression model ) ⋯ ( ∼ (, Σ) ) ⋯ ( , ⋮ ⋱ ⋮ estimators estimators are normally distributed ( ( Σ How to to assessHow our of the precision estimates? hold, A1–A5and the errors are normally distributed If (A6),then the LS The varianceLS of estimator is: Applied Econometrics-QEM so we Page 38 Page … so so that: 2 i ˆ e 1 = .943 N − i 23.874 NK the variance the ∑ = Regression model 2

ˆ σ 1718 75 3 1718.943 2 i ˆ e 1 is is the number of degrees of freedom = − − 75 i 23.874 4.8861 NK ∑ = = = = = 2 ˆ N ˆ σ σ However, we don’tHowever, know need to substitute it with the unbiased estimator: where sales model For Applied Econometrics-QEM Page 39 Page 68 . 0 1.096 ∑ .3 6.35 47 . 1.20 0 40 Regression model .47 Σ 1.20 0.02 40.3 6.80 0.75 Now we are weare Now ready to calculate the precision of estimates with the feasible formula: The standard The standard errors are: For the the For sales model wehave: Σ Applied Econometrics-QEM Page 40 Page Regression model Table 3.1 Least Squares Estimates from 10 Random Squares Random Samples 3.1LeastEstimates10 from Table Monte Carlo experiment:Monte

Applied Applied Econometrics QEM Page 41 Page ∼ with its estimate changes the changes itswithestimate -Student,sothat: t k ) b Regression model ( IMPORTANT!!! of thevariance Replacing distributionfrom normal to Applied Econometrics-QEM Page 42 Page (1 )% 100 2 ,1 = k for

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β k b () − k se b = ( t ± interval estimateof interval In general,In then:holdA1-A6 if The Applied Econometrics-QEM ) 5.723 Page 43 Page ] an increase in increase an ) 1 ( 1 ) ( , , ≤ ≤ )=72]: Regression model N-K ≤ ) ≤ ( , , model we we [( havemodel 1.993 se( ) 1.993 se( ) .95 − × ≤β≤+ × = 2 222 2 SALES SALES [ 7.9079 1.993 1.096, 7.9079 1.993 1.096 10.093, −−×− +×= − Pb b b b ( For to will lead $1by price decreasing Interpretation: $10,093.and $5,723 between somewhere revenue Interval Interval estimation: Applied Econometrics-QEM Page 44 Page

, c 2c Regression model we we have: c ∼ / √ Distribution for the linear Distributionfor combination of parameters obtainwish tomay We distribution the of combination a for linear parameters: where c1 and c2 are constants that we specify we thatconstantsare and c2 where c1 Then Applied Econometrics-QEM ) n $3,471 n Page 45 Page 0.7096 ,5.835

0.8 ( ) 0.4 7.91 0.8 × 1.86 4.6532 Regression model 0.4 × 0.8 0.16 × 1.2 0.64 × 0.47 0.64 × (0.02) −× +× = 0.4 4.6532 1.666 0.7096,4.6532 1.666 0.7096 3.471 ( The estimator is: estimator The 90% interval: The betwee lie sales willin increase expected the that Indicates probability 90% and $5,835 with Example: dropprice theby $800 and advertisingincreaseSuppose want we to salesis: in change expectedThe by 40 cents. Applied Econometrics-QEM Page 46 Page Regression model Hypothesis Tests Hypothesis

Applied Applied Econometrics QEM e e ter a a Page 47 Page Regression model consists of4 steps: ę the sample:LS itsestimate andstandard error 1. Setting H0 and H1 2. Calculate astatistic test 3. Calculate aregion rejection 4. conclusion A about a population theinformationabouta intocontained ofsample data Hypothesis testing = comparison conjecture of we hav a Ineconometric modelsrepresented hypothesesare as parameters aboutmodel statements usetheinformationHypothesis testsabouta parame The procedur from

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Applied Applied Econometrics QEM n) Page 62 Page out-of-sample better decisionsbetter Regression model ‰ business (e.g. forecasts ofsales) policywhomakers (e.g. forecast ofoutput, inflatio – – Predictioninference = about observations to: is importantpredictto ability The Accurate predictionsAccurate Applied Econometrics-QEM ) Page 63 Page ( =0): t Regression model comes comes from the fitted regression 0 We would like the forecast error to be small, implying We that forecast our close is to the value weare predicting The LS predictor The predictor LS of y lineassume (we that predition is for Let us define the forecast error: Applied Econometrics-QEM Eq.4.2 ) ) Page 64 Page ̅ ) ( ̅ 0 ∑ 1 ) (( (unbiased forecast): 1 Regression model ( 0 error The variance the of forecast is The expected value of Two sources of forecast variance: - random - estimation error Applied Econometrics-QEM )] Page 65 Page ) )[′( )′ )′() ( ( Regression model ( (exogenous vars. error)

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Applied Applied Econometrics QEM nthe Page 69 Page 2 – kurtosis 3 K − Jarque–Bera test K () ) 2 Regression model 2 (      – skewness, 6 4 N S = + ∼ JB S - size, sample N a histogram formal e.g. test, statistical – – assumption that theerrors assumptionthat are normallydistributed this check can We using: Hypothesis tests and interval estimates often andestimates Hypothesisintervaltests relyo Undernull, the Applied Econometrics-QEM Page 70 Page Regression model expenditure example expenditure

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Applied Applied Econometrics QEM 0 joint 2 Page 73 Page ed a ed 0 nonzero bothare or Regression model 0 β thenulltheinnotrestrictionshave β 0, 0 or : assumes theassumesH parameterin:restrictions = = ≠ ≠ β β : : 12 3 4 1 2 For example,for model the 0 3 4 1 3 4 ββ β β β β H H =+ + + + = + + SALES PRICE ADVERT ADVERT e SALES PRICE e aretrue, i.e.: hypothesis. possibleajoint hypothesisbe: could model: Unrestricted theimposedbeenonmodel Restrictedmodel A null hypothesis withcallnullconjectures ismultiplehypothesis A Applied Econometrics-QEM N N sums sums of hasthe and the and Page 74 Page F U SSE ) − , , thestatistic then () − Regression model U R U numeratorof freedom degrees and -thenumber ofrestrictions) SSE N K SSE SSE J J ∼ (, ) ( ( J R = F SSE denominator degreesdenominatorof freedom -distributionwith -testfor hypothesis:a comparisonthejoint of the squarederrors from model theunrestricted onerestricted F - K Ifthe nullhypothesis istrue F Applied Econometrics-QEM 8.44 Page 75 Page ) () > 8.44) = 0.0005 = 8.44)> 1532.084 75 4 = 3.126 we reject the null rejectwethe3.126 = Regression model (2, 71) (2, 1896.391 1532.084 2 ( F ( 2,71 c, P F = ) p − − : advertising does have a significanthaveadoes advertising : () − − , continuation: , U R U = 8.44> = F -value is is -value SSE N K SSE SSE J p ( = = = F Example Since The Conclusion salesupon revenue effect Applied Econometrics-QEM K Page 76 Page = K K L is nonzero for for isnonzero 2,3, 0 k β 1 K β Regression model = β + i i K y e 0, , β 1 22 33 0, ββ β β = = = = + + + + + y xx xe β : : the of 0 2 3 1 H H At least one k K

Overall significance test of the regression model regression the testof significance Overall model the For examine: we is: modelrestrictedThe Applied Econometrics-QEM Page 77 Page () ∼ ∼ (, ) )/ : Regression model /( ) ( -statistic of the Wald test: Wald -statistic of the F Lagrange Multipliertest: Comparison and tests LM ofF The Given the LS estimator Applied Econometrics-QEM i 2 Page 78 Page p Regression model β =β = : 0 16.88 -value .0002 0 3 4 8.44 -value .0005 = = H 2 F p χ = = 12 3 4 =β+β +β +β + SALES PRICE ADVERT ADVERT e

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