Data Considerations and Ordinary Least Squares Estimation of Single-Equation Econometric Models

Transcription

1 Chapter Data Consderatons and Ordnary Least Squares Estmaton of Sngle-Equaton Econometrc Models

2 Secton. Data

3 Crtcal Ingredent n All Appled Econometrc Models Suffcently large amount of hstorcal data. Ask not what you can do to the data, but rather what the data can do for you. Data Types Tme-Seres Cross-Sectonal Combnaton (Panel) 3

4 Crtcal Ingredent Crtcal Ingredent data (sample suffcently large ) Tme-seres data daly, weekly, monthly quarterly, annual DAILY closng prces of stock prces WEEKLY measures of money supply MONTHLY housng starts QUARTERLY GDP fgures ANNUAL salary fgures Cross-Sectonal Data Snapshot of actvty at a gven pont n tme Survey of household expendture patterns Sales fgures from a number of supermarkets at a gven pont n tme Pooled Tme-Seres, Cross-Sectonal Data 4

5 Quote from Lord Kelvn I often say that when you can measure what you are speakng about, and express t n numbers, you know somethng about t; but when you cannot measure t, when you cannot express t n numbers, your knowledge s of a meager and unsatsfactory knd. 5

6 Other Notable Quotes Concernng Data n Appled Econometrcs R.H. Coase: If you torture the data long enough, nature wll confess. E.E. Leamer (983): There are two thngs you are better off not watchng n the makng: sausages and econometrc estmates. 6

7 Secton. Gettng a Feel for the Data

8 Examnng the Data Get a feel for the data. Plots of key varables Scatter plots Descrptve statstcs Mean Outler(s) Medan Skewness Mnmum Kurtoss Maxmum Dstrbuton Standard devaton 8

9 Descrptve Statstcs X Let X X correspond to a vector of T observatons for M X T the varable x. The mean, a measure of central tendency, corresponds to the average of the set of observatons correspondng to a partcular data seres. The mean s gven by. x T The unts assocated wth the mean are the same as the unts of x,,,, T. T x 9

10 Descrptve Statstcs The medan also s a measure of central tendency of a data seres. The medan corresponds to the 50 th percentle of the data seres. The unts assocated wth the medan are the same as the unts of x,,,, T. To fnd the medan, arrange the observatons n ncreasng order. When sample values are arranged n ths fashon, they often are called the st, nd, 3 rd, order statstcs. In general, f T s an odd number, the medan s the order statstc whose number s T and T +. If T s an even number, the medan corresponds to the average of the order statstcs whose numbers are T +. 0 contnued...

11 Descrptve Statstcs The standard devaton s a measure of the spread or dsperson of a seres about the mean. The standard devaton s gven by S The unts assocated wth the standard devaton are the same as the unts of x. T ( x T x) The varance also s a measure of the spread or dsperson of a seres about the mean. ˆ The varance s expressed as Note that σ The unts assocated wth the varance are the square of the unts of x. contnued... T ( x T x).. ˆ σ S.

12 Descrptve Statstcs The mnmum of a seres corresponds to the smallest value, mn(x, x,, x T ). The unts assocated wth the mnmum are the same as the unts of x. The maxmum of a seres corresponds to the largest value, max(x, x,, x T ). The unts assocated wth the mnmum are the same as the unts of x. The range of a seres s the dfference between the maxmum and the mnmum values. The range s expressed as Range x max x - mn x. The unts assocated wth the range are the same as the unts of x. contnued...

13 Descrptve Statstcs Skewness s a measure of the amount of asymmetry n the dstrbuton of a seres. If a dstrbuton s symmetrc, skewness equals zero. If the skewness coeffcent s negatve (postve), then the dstrbuton of the seres has a left (rght) tal. The greater the absolute value of the skewness statstc, the more asymmetrc s the dstrbuton. The skewness coeffcent s gven by T ( x ˆ 3 m T S x) 3. The skewness statstc s a untless measure. 3 contnued...

14 Descrptve Statstcs Kurtoss s a measure of the flatness or peakedness of the dstrbuton of a seres relatve to that of a normal dstrbuton. A normal random varable has a kurtoss of 3. A kurtoss statstc greater than 3 ndcates a more peaked dstrbuton than the normal dstrbuton. A kurtoss statstc less than 3 ndcates a more flat dstrbuton than the normal dstrbuton. The kurtoss coeffcent s gven by kˆ T T ( x S 4 x) 4. The kurtoss coeffcent also s a untless measure. 4

15 Descrptve Statstcs The Jarque-Bera (JB) test statstc combnes the skewness and kurtoss coeffcents to produce another measure of the departure of normalty of a seres. Ths T measure s gven by ˆ ( ˆ JB m + k 3). 6 4 For a normal dstrbuton, m ˆ 0 and kˆ 3. Thus, the JB statstc s 0 for normal dstrbutons. Values greater than 0 ndcate the degree of departure from normalty. (Jarque and Bera, 980) 5 contnued...

16 Descrptve Statstcs The coeffcent of varaton s the rato of the standard devaton to ts mean. Ths measure typcally s converted to a percentage by multplyng the rato by 00. Ths statstc descrbes how much dsperson exsts n a seres relatve to ts mean. Ths measure s gven by CV S 00%. The utlty of ths nformaton s that x n most seres the mean and standard devaton change together. As well, ths statstc s not dependent on unts of measurement. 6

17 7 Correlaton Coeffcent The correlaton coeffcent s a measure of the degree of lnear assocaton between two varables. The statstc, denoted by r, s gven by r ( x T T ( x x) T x)( y y) ( y y). Whle r s a pure number wthout unts, r always les between - and +. Postve values of r ndcate a tendency of x and y to move together, that s, large values of x are assocated wth large values of y and small values of x are lnked wth small values of y. When r s negatve, large values of x are assocated wth small values of y and small values of x are assocated wth large values of y. The closer to +, the greater the degree of drect lnear relatonshp s between x and y. The closer to -, the greater the degree of nverse lnear relatonshp s between x and y. Fnally, when r 0, there s no lnear assocaton between x and y.

18 Descrptve Statstcs The mode corresponds to the most frequent observaton n the data seres x, x,, x r. The unts assocated wth mode are the same as the unts of x. In emprcal applcatons, often the observatons are non-repettve. Hence, ths measure often s of lmted usefulness. 8

19 Data Example Prces and quanttes sold of Prego Spaghett Sauce by week obs PPRG QPRG 6/03/ /0/ /7/ /4/ /0/ /08/ /5/ // /9/ /05/ // /9/ /6/ contnued...

20 obs PPRG QPRG 9/0/ /09/ /6/ /3/ /30/ /07/ /4/ // /8/ /04/ // /8/ /5/ /0/ /09/ /6/ /3/ /30/ /06/ /3/ obs PPRG QPRG /0/ /7/ /03/ /0/ /7/ /4/ /0/ /09/ /6/ /3/ /30/ /06/ /3/ /0/ /7/ /04/ // /8/ /5/ /0/99 NA NA 0

21 Tme-Seres Plot Volume of Prego Spaghett Sauce Sold by Week Q3 99Q4 99Q 99Q QPRG

22 Descrptve Statstcs and Hstogram Descrptve Statstcs and the Hstogram Assocated wth the Volume of Prego Spaghett Sauce Seres: QPRG Sample 6/03/99 6/0/99 Observatons 5 Mean Medan Maxmum Mnmum Std. Dev Skewness Kurtoss Jarque-Bera Probablty 0.974

23 Tme-Seres Plot.95 Prce of Prego Spaghett Sauce by Week Q3 99Q4 99Q 99Q 3 PPRG

24 Descrptve Statstcs Prce of Prego Spaghett Sauce Seres: PPRG Sample 6/03/99 6/0/99 Observatons 5 Mean Medan Maxmum.9367 Mnmum.6568 Std. Dev Skewness Kurtoss Jarque-Bera 9.98 Probablty

25 PPRG versus QPRG Weekly Scatter Plot of Prces and Quanttes Sold of Prego Spaghett Sauce PPRG QPRG

26 Correlaton Matrx The correlaton between the prce and quantty sold of Prego Spaghett Sauce s PPRG QPRG PPRG QPRG

27 The MEANS Procedure Varable N Mean Std Dev Mnmum Maxmum QPREGO PPREGO The UNIVARIATE Procedure Varable: QPREGO Moments 7 N 5 Sum Weghts 5 Mean Sum Observatons Std Devaton Varance Skewness Kurtoss Uncorrected SS Corrected SS Coeff Varaton Std Error Mean.4693

28 Basc Statstcal Measures Locaton Varablty Mean Std Devaton Medan Varance Mode. Range Interquartle Range Tests for Normalty Test --Statstc p Value Shapro-Wlk W Pr < W Kolmogorov-Smrnov D Pr > D 0.37 Cramer-von Mses W-Sq Pr > W-Sq Anderson-Darlng A-Sq Pr > A-Sq

29 Quantles (Defnton 5) Quantle Estmate 00% Max % % % % Q % Medan % Q.06 0% % % % Mn Extreme Observatons Lowest Hghest----- Value Obs Value Obs

30 The UNIVARIATE Procedure Varable: PPREGO Moments N 5 Sum Weghts 5 Mean Sum Observatons Std Devaton Varance Skewness Kurtoss Uncorrected SS Corrected SS Coeff Varaton Std Error Mean Basc Statstcal Measures Locaton Varablty Mean Std Devaton Medan Varance Mode. Range Interquartle Range

31 Tests for Normalty Test --Statstc p Value Shapro-Wlk W Pr < W Kolmogorov-Smrnov D Pr > D 0.75 Cramer-von Mses W-Sq Pr > W-Sq Anderson-Darlng A-Sq Pr > A-Sq Quantles (Defnton 5) Quantle Estmate 00% Max % % % % Q % Medan % Q.8 0% % % % Mn.6563

32 Secton.3 Massagng the Data

33 Massagng the Data Ask not what you can do to the data, but rather what the data can do for you. Express data n real or nomnal terms. Total or per capta measures Levels, changes, or percentage changes Smoothng Movng averages Exponental smoothng Seasonal adjustments Imputatons for mssng data See also Gerhard Svolba, Data Preparaton for Analyss Usng SAS (006). 33

34 Smoothng Smoothng technques provde a means of removng or at least reducng volatle short-term fluctuatons n a tme seres. These technques can be useful because t s often easer to dscern trends and cyclcal patterns and otherwse analyze a smoothed seres. Seasonal adjustment s really a specal form of smoothng. 34

35 N-Perod Movng Averages t n t t t t X X X X g e X X X n X )... (.. )... ( t t t t t X X X X g e )... (.. Use of the EXPAND Procedure

36 EWMA: Exponentally Weghted Movng Average (Exponental Smoothng) X X t t α. αxt + α( α)xt + α( α) Xt 0 < α <.X t +.09X t +.08X t X 3 t u u t t α.9 X t.9x t +.09X t +.009X t u t Weghts must sum to. Use of the EXPAND Procedure 36

37 A Plot of the Actual Seres FUBK and Its -Month Movng Average 80,000 70,000 60,000 50,000 40,000 30,000 0,000 0, FUBK FUBKMOVAVG

38 The MEANS Procedure Varable N Mean Std Dev Mnmum Maxmum fubk fubkmov The UNIVARIATE Procedure Varable: fubk 38 Moments N 07 Sum Weghts 07 Mean Sum Observatons Std Devaton Varance Skewness Kurtoss Uncorrected SS.7334E Corrected SS E0 Coeff Varaton Std Error Mean

39 Basc Statstcal Measures Locaton Varablty Mean Std Devaton 6986 Medan Varance Mode. Range Interquartle Range 5730 Tests for Normalty Test --Statstc p Value Shapro-Wlk W Pr < W <0.000 Kolmogorov-Smrnov D Pr > D <0.000 Cramer-von Mses W-Sq Pr > W-Sq < Anderson-Darlng A-Sq Pr > A-Sq <

40 Quantles (Defnton 5) Quantle Estmate 00% Max % % % % Q % Medan % Q 470 0% 397 5% 3667 % % Mn 3380 Extreme Observatons ----Lowest Hghest---- Value Obs Value Obs

41 Mssng Values -----Percent Of----- Mssng Mssng Value Count All Obs Obs The CORR Procedure 3 Varables: fubk fubkmov debtncome Smple Statstcs Varable N Mean Std Dev Sum Mnmum Maxmum fubk fubkmov debtncome

42 Pearson Correlaton Coeffcents Prob > r under H0: Rho0 Number of Observatons fubk fubkmov debtncome fubk <.000 < fubkmov <.000 < debtncome <.000 <

43 The Estmate of Alpha n the Exponental Smoothng Process Sample: 99M07 000M05 Included observatons: 07 Method: Sngle Exponental Orgnal Seres: FUBK Forecast Seres: FUBKSES Parameters: Alpha Sum of Squared Resduals.95E+09 Root Mean Squared Error End of Perod Levels: Mean

44 A Plot of the Actual Seres FUBK and the Exponental Smoothng of Ths Seres 80,000 70,000 60,000 50,000 40,000 30,000 0,000 0, FUBK FUBKSES

45 Seasonal Adjustment X or X Bureau of the Census Procedure Apply the seasonal adjustment technque to a seres for monthly housng starts. To do ths, compute a -month average ( Ŷ t ) of the orgnal seres Y t and then dvde (Y t ) by Y ˆ ( / ˆ ). Notce that Z contans t for example, Z t Y t Y t t (roughly) the seasonal and rregular components of the orgnal seres. Remove the rregular component by averagng the values of Z t that correspond to the same month, that s, compute Z ~ Z ~,...Z ~,. Then compute the fnal seasonal ndces Z, Z,... Z by multplyng Z ~, Z ~,... Z ~ by a factor that brngs the sum to. 45

46 Seasonal Adjustment Example Monthly Per Capta Consumpton of Orange Juce 989:0 to 00:09 MONTH INDEX MONTH INDEX January.397 July February August March September Aprl October.0360 May November June December Sum Use of PROC X or PROC X 46

47 Plot of the Actual Seres and the Seasonally Adjusted Seres A Plot of the Actual Seres Called ALLOJGALPC and Its Seasonally Adjusted (SA) Verson ALLOJGALPCSA ALLOJGALPC

48 Addtonal Transformatons of Varables Trgonometrc (cycles) Logarthmc (elastctes, flexbltes) Polynomal (economes of scale) Lags n varables The logarthmc transformaton mght lessen the possblty of heteroscedastcty ssues. The use of lags mght lessen the possblty of seral correlaton ssues. Be careful of the use of polynomals and the potental dangers of over-fttng. 48

49 Secton.4 Estmaton of the Smple Lnear Regresson Model

50 The Smple Lnear Regresson Model Dependent Varable Left-Hand Sde Varable Explaned Varable Regressand Response Varable Independent Varable Rght-Hand Sde Varable Explanatory Varable Regressor Control Varable y β 0 + β x + u Coeffcents β 0 : Intercepts β : Slope Error Term Dsturbance Term Innovaton 50

51 Y (Depend dent Varable) Graphcal Illustraton y β 0 + β x + u Regresson lne: E(y x) β 0 + β x β : Slope β 0 5 How do you nterpret coeffcents? x x+h X (Independent Varable)

52 An Example: Relatonshp between Total Personal Bankruptces and Real GDP Queston: What s the effect of real gross domestc product (GDP) on total personal bankruptces? Known nformaton: Dependent varable: total personal bankruptces (TPB) Explanatory varable: real GDP (RGDP) Regresson: TPB β 0 + β RGDP + u Interpretaton: β measures the change n real GDP on TPB. β 0 represents the autonomous level of total personal bankruptces. 5

53 Random Samplng Randomly sample n observatons from a populaton (979:4 to 003:). 94 quarterly observatons For each observaton, TPB t β 0 + β RGDP t + u t Goal: Estmate β 0 and β Explanatory Varable RGDP Dependent Varable TPB RGDP 979:4 TPB 979:4 RGDP 980: TPB 980: RGDP 003: TPB 003: 53

54 Ordnary Least Squares, Regresson Lne, Ftted Values, and Resduals y y 4 y. 3 } û. 3 û { y y. } û û 4 {. yˆ ˆ β + ˆ βx OLS: choose β 0 and β to mnmze these sum of squared predcton errors. 0 x x x 3 x 4 x 54

55 ,800,000,600,000,400,000,00,000 TPB,000, , , ,000 00, ,000 6,000 8,000 0,000, RGDP

56 Intutve Thnkng about Ordnary Least Squares Ordnary least squares (OLS) s fttng a lne through the sample ponts so that the sum of squared predcton errors s as small as possble; hence the term least squares. Resdual û s an estmate of the error term, u, and s the dfference between sample pont (actual value) and the ftted lne (sample regresson lne). uˆ AV FV,,..., n Actual Value Ftted Value 56

57 Mnmzng Resdual Sum of Squares ( ) 0 ˆ ˆ 0 n x y β β ( ) ( ) n n x y u 0,, mn mn 0 0 ˆ β β β β β β ) ) ) ) ) ) ( )( ) ( ) n n y y x x ˆβ Frst order condtons: ( ) 0 ˆ ˆ 0 0 n x y x β β ( ) x x y x 0 ˆ ˆ β β Interpretaton: The slope estmate s the sample covarance between x and y dvded by the sample varance of x.

58 The Resdual Varance Use the resdual to estmate the resdual varance. Ths varance represents the amount of dsperson about the ftted model. ( ) ˆ ˆ ˆ ˆ ˆ ˆ x u x u x y u β β β β β β û SSE s the resdual or error sum of squares, and (n-) s the degrees-of-freedom ( ) ( ) ( ) ( ) / ˆ ˆ ˆ ˆ ˆ 0 0 n SSE u n x u u σ σ β β β β s of an unbased estmator Then,

59 Standard Error of OLS Estmates ˆ σ ˆ σ standard error of the regresson The standard error of β s gven by se ( ˆ β ). se ( ˆ β ) ˆ σ /( ( x x ) ( ) ) The standard error of β s gven by se ( ˆ β ) se ( ˆ β ) 0 ( ˆ σ x x n x ( ) )

60 Goodness-of-Ft: Coeffcent of Determnaton Goodness-of-ft: How well does the lnear regresson model explan the varaton nherent n the dependent varable? Y Yˆ Y ˆ Yˆ Y Unexplaned varaton ( Y Y ˆ ) u Explaned varaton ( Yˆ Y ) 60 X X X

61 y Goodness-of-Ft: Some Termnology Goodness-of-ft: How well does the smple regresson lne ft the sample data? Calculate R SSR/SST SSE/SST. yˆ + uˆ Then defne the followng: Thus, ( y ) y ( yˆ ) y ( y yˆ ) s the total sum of s uˆ SST SSR + SSE the regresson sum of squares (SST). squares (SSR). s the resdual or error sum of squares (SSE). 6 contnued...

62 Goodness-of-Ft Concept: measures the proporton of the varaton n the dependent varable explaned by the regresson equaton. Formula: R Explaned sample varablty Total sample varablty n ( yˆ y ) ˆ SSR SST SSE n ( y y) SST Range: between zero and one Example: R 0.78, the regresson equaton explans 78% of the varaton n y. 6

63 The R-square Statstc and the Adjusted R ( yˆ ) R y Explaned sample varablty R n Total sample varablty ( y y) n SSR SST SSE SST Adjusted R SSE /( n k ) R SST /( n ) Questons: (a) Why do you care about the adjusted R? (b) Is adjusted R always better than R? (c) What s the relatonshp between R and adjusted R? 63

64 Secton.5 SAS Output of the Smple Lnear Regresson of Total Personal Bankruptces (TPB) on Real Gross Domestc Product (RGDP)

65 The REG Procedure Model: MODEL Dependent Varable: TPB Number of Observatons Read 94 Number of Observatons Used 94 Analyss of Varance p-value Sum of Mean Source DF Squares Square F Value Pr > F Model.58735E E <.000 Error 9.079E Corrected Total E3 Estmate of resdual varance Root MSE 0794 R-Square Dependent Mean Adj R-Sq Coeff Var Goodness-of-Ft statstcs

66 Parameter Estmates Parameter Standard Varable DF Estmate Error t Value Pr > t Intercept <.000 RGDP <.000 Covarance of Estmates p-value Varable Intercept RGDP Intercept RGDP

67 Smple Lnear Regresson of RGDP on Total U.S. Personal Bankruptces σˆ se ( ˆ β 0 se ( ˆ β ) SSE 9,65,39,045 ) 53,44 R R SST SSR SSE.079 0

68 Actual Values, Ftted Values, and Resduals Obs TPB predtpb restpbrgdp

69 Predctng Total U.S. Personal Bankruptces: 003:, 003:3, and 003:4 Let t 003: RGDP 003: 0,47.8 TPB 003:,58,734 t 003:3 RGDP 003:3 0,580.7 TPB 003:3,60,70 t 003:4 RGDP 003:4 0,697.5 TPB 003:4,640,974 T PB ˆ t RGDP t Key Pont: Need nformaton on out-of-sample values for RGDP n order to predct out-of-sample values for TPB. Condtonal Forecastng 69

70 Secton.6 Estmaton of the Multple Regresson Model The Generc Sngle-Equaton Econometrc Model

71 Multple Regresson Model y β 0 + β x + β x + + β k x k + u β 0 s stll the ntercept. β to β k are all called slope parameters. u s stll the error term (or dsturbance term). You can stll mnmze the sum of squared resduals. 7

72 Random Samplng Dependent Varable Prce Income Advertsng Q P I A Q P I A Q 3 P 3 I 3 A 3 Q 3 P 3 I 3 A 3 Use OLS to estmate the coeffcents to mnmze the sum of squared errors. ) β n n ( ) uˆ ( ) y β0 βp βi β3a mn mn ) ) ) ) ) ),,,,,, 0 β β β 3 β 0 β β ) β 3 ) ) ) ) 7

73 OLS Estmates Assocated wth the Multple Regresson Model ) ( y )) ( ( ˆ ) ( nx y y k nx x x x x x x x y x x x k k T T + β L L ) ) (( ˆ ˆ ˆ ˆ ) ) x (( the transpose of s ) ( y )) ( ( 0 x k xn k x nx y k nx x x x x k T n kn n n β β β β M M O M M M

74 Goodness-of-Ft y yˆ + uˆ Defntons : ( ) y y s the total sum of squares (SST). ( yˆ ) y s the regresson sum of squares (SSR). uˆ s the resdual(or error)sum of squares (SSE). Then SST SSR + SSE. 74 contnued...

75 Sample Regresson Lne and Sample Data How well does your sample regresson lne ft your sample data? The R-square statstc of regresson s the fracton of the total sum of squares (SST) that s explaned by the model. R SSR/SST SSE/SST 75

76 More about the R-square Statstc R can never decrease when another explanatory varable s added to a regresson, and usually t wll ncrease. Because R wll usually ncrease (or at least not decrease) wth ncreases n the number of explanatory varables, t s not a good way to compare alternatve models wth the same dependent varable. 76

77 Estmatng the Resdual Varance df n (k + ), or df n k df (that s, degrees of freedom) s (number of observatons) (number of estmated parameters) ( ) u ( n k ) SSE df s σˆ ˆ 77

78 Varances and Covarances of OLS Parameter Estmates Varance: Covarance Matrx of OLS Parameter Estmates ( ˆ) s (x T x) - Var β Ths matrx s a functon of the estmated resdual varance. 78

79 The R-square Statstc and the Adjusted R ( yˆ ) R y Explaned sample varablty R n Total sample varablty ( y y) n SSR SST SSE SST Adjusted R SSE /( n k ) R SST /( n ) Questons: (a) Why do you care about the adjusted R? (b) Is adjusted R always better than R? (c) What s the relatonshp between R and adjusted R? 79

80 80 Model Selecton Crtera T T p s T T MSE s the penalty factor. p T t T e t T t T e t T T Akake Informaton Crteron (AIC) AIC MSE MSE p Useful way of dscernng alternatve model specfcatons. For each specfcaton, calculate the assocated model selecton statstcs. Then the smallest statstc yelds the most approprate model specfcaton. Schwarz or Bayesan Informaton Crteron (SIC) or (BIC) SIC ( BIC p s the number of parameters to be estmated. T s the number of observatons. ) e p T T p T MSE

81 Model Selecton Crtera Example Model Model Model 3 AIC SIC Whch model to choose? 8

82 Secton.7 Example: SAS Output of the Demand Functon for Shrmp

83 Example: SAS Output of the Demand Functon The REG Procedure Model: MODEL Dependent Varable: QSHRIMP Number of Observatons Read 97 Number of Observatons Used 97 Quantty sold of shrmp Analyss of Varance Sum of Mean Source DF Squares Square F Value Pr > F Model <.000 Error Corrected Total Root MSE 4.93 R-Square Dependent Mean Adj R-Sq Coeff Var contnued...

84 Parameter Estmates Parameter Standard 95% Varable DF Estmate Error t Value Pr > t Confdence Intercept PSHRIMP < PFIN PSHELL ADSHRIMP ADFIN ADSHELL Prce of shrmp Prce of fnfsh Prce of other shellfsh Advertsng for shrmp Advertsng for fnfsh Advertsng for other shellfsh

85 Correlaton of Qshrmp t and predqshrimp t Varables: QSHRIMP predqshrimp Smple Statstcs Varable N Mean Std Dev Sum Mnmum Maxmum QSHRIMP predqshrimp Smple Statstcs Varable Label QSHRIMP predqshrimp Predcted Value of QSHRIMP Pearson Correlaton Coeffcents, N 97 Prob > r under H0: Rho0 QSHRIMP pred QSHRIMP QSHRIMP < predqshrimp Predcted Value of QSHRIMP <.000

86 Varance-Covarance Matrx of Estmated Coeffcents of the Econometrc Model Varable Intercept PSHRIMP PFIN PSHELL Intercept PSHRIMP E E-6 PFIN E E-6 PSHELL E E ADSHRIMP E E E-6 ADFIN E E E-7 ADSHELL E Varable ADSHRIMP ADFIN ADSHELL Intercept PSHRIMP E E PFIN E E E-6 PSHELL E E ADSHRIMP ADFIN E-6 ADSHELL E

87 Model Selecton Crtera for the QSHRIMP Problem MSE SSE/T / s T T-P MSE (6.9) 7.56 AIC e P T MSE e 4 97 (6.9) 8.8 SIC T P T MSE (6.9).67 87