Specification. Hypothesis tests and confidence intervals

SINGLE REGRESSION Specification, Hypothesis Tests and Confidence Intervals Specification of Equations
Excel: Data  Data Analysis
EViews: Object  New object  Equation V. Ozolina Econometrics Specification of Equations
Excel: Data  Data Analysis  Regression V. Ozolina Econometrics Specification of Equations
Excel: Data  Data Analysis  Regression SUMMARY OUTPUT Regression Statistics Multiple R 0.796991 R Square 0.635195 Adjusted R Square 0.63105 Standard Error 3118.862 Observations 90 ANOVA df SS 1.49E+09 8.56E+08 2.35E+09 MS 1490464776 9727299.3 Coefficients Standard Error 3313.83 1560.228 64.7576 5.231497 t Stat 2.12393914 12.3784059 Regression Residual Total Intercept W_NET_SA 1 88 89 F Significance F 153.2249 5.69E-21 P-value 0.036481 5.69E-21 Lower 95% 213.2041 54.36109 RESIDUAL OUTPUT Observation Predicted IMP_FARMAC_SA 1 13607.51 2 13894.03 Residuals -2278.63 -3737.32 V. Ozolina Econometrics Upper 95% Lower 95.0% Upper 95.0% 6414.456 213.2041 6414.456 75.1541 54.36109 75.1541 Testing if OLS Assumptions Hold



A1: E(ui) = 0 Average value of the error term is zero – it is important, because we do not know the exact errors A1 will always hold, if we use constant term in the equation If we do not use the constant term in the equation, we cannot test, if A1 holds!!! V. Ozolina Econometrics Testing if OLS Assumptions Hold
A2: Var(ui) = σ2 < ∞ Homoscedasticity
The opposite situation - heteroscedasticity
V. Ozolina Econometrics Testing if OLS Assumptions Hold
Var(ui) = σ2 < ∞ - homoscedasticity
Problems:
Property 2 does not hold – OLS estimates of the coefficients are no longer BLUE – coefficient estimates are not biased, but they are not efficient (do not have minimum variance),
Standard error values may be inaccurate (depending on the form of heteroscedasticity – standard errors are overvalued or undervalued)  standard errors are not usable!!! V. Ozolina Econometrics Testing if OLS Assumptions Hold
Var(ui) = σ2 < ∞ - homoscedasticity
How to test:
Graphical analysis – correlation diagram of the error terms and forecasts
Using White’s test
H0: assumption holds, Ha: does not hold > 0.05 → 
 

−   < 0.05 → 
    V. Ozolina Econometrics Testing if OLS Assumptions Hold: Example
Excel SUMMARY OUTPUT Regression Statistics Multiple R 0.796991 R Square 0.635195 Adjusted R Square 0.63105 Standard Error 3118.862 Observations 90 ANOVA df Regression Residual Total SS 1.49E+09 8.56E+08 2.35E+09 MS 1490464776 9727299.3 Coefficients Standard Error 3313.83 1560.228 64.7576 5.231497 t Stat 2.12393914 12.3784059 1 88 89 Intercept W_NET_SA F Significance F 153.2249 5.69E-21 P-value 0.036481 5.69E-21 Lower 95% 213.2041 54.36109 RESIDUAL OUTPUT Observation Predicted IMP_FARMAC_SA 1 13607.51 2 13894.03 Residuals -2278.63 -3737.32 V. Ozolina Econometrics Upper 95% Lower 95.0% Upper 95.0% 6414.456 213.2041 6414.456 75.1541 54.36109 75.1541 Testing if OLS Assumptions Hold: Example
Graphical analysis in Excel
Is the spread of the error term increasing or decreasing? V. Ozolina Econometrics Testing if OLS Assumptions Hold: Example
White’s test in EViews V. Ozolina Econometrics Testing if OLS Assumptions Hold: Example p-value = 0.14
0.14 > 0.05  H0 is accepted  Error term is homoscedastic
White Heteroskedasticity Test: F-statistic Obs*R-squared 1.972629 3.904253 Prob. F(2,87) Prob. Chi-Square(2) 0.145263 0.141972 Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 10/06/15 Time: 13:50 Sample: 2005M01 2012M06 Included observations: 90 Variable Coefficient Std. Error t-Statistic Prob. C W_NET_SA W_NET_SA^2 4448473. -44395.65 202.4366 47843715 380423.2 715.4161 0.092979 -0.116701 0.282963 0.9261 0.9074 0.7779 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.043381 0.021389 19064387 3.16E+16 -1634.879 2.241216 V. Ozolina Econometrics Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 9511137. 19271604 36.39731 36.48063 1.972629 0.145263 Testing if OLS Assumptions Hold
Var(ui) = σ2 < ∞ - homoscedasticity
What to do, if the assumption does not hold:
Transform the data, for example, using logarithms V. Ozolina Econometrics Testing if OLS Assumptions Hold
Var(ui) = σ2 < ∞ - homoscedasticity
What to do, if the assumption does not hold:
If the heteroscedasticity form is known, we can use GLS (Generalized Least Squares)
If there are a lot of data, but the solution is not known, we can use HAC (Heteroscedasticity consistent coefficient covariance) V. Ozolina Econometrics Testing if OLS Assumptions Hold

A3: Cov(ui,uj) = 0 There is no autocorrelation (spatial correlation) in the error term V. Ozolina Econometrics Testing if OLS Assumptions Hold
Cov(ui,uj) = 0
Problems: - no autocorrelation Property 2 does not hold – OLS estimates of the coefficients are no longer BLUE – coefficient estimates are not biased, but they are not efficient even (do not have minimum variance) if the sample size is large,
Standard error values may be inaccurate  standard errors are not usable!!!
In case of the positive autocorrelation, the value of R2 is overstated
V. Ozolina Econometrics Testing if OLS Assumptions Hold
Cov(ui,uj) = 0 - no autocorrelation
How to test:
Graphical analysis  Time-sequence plot – residuals against time  Plot residuals against their values lagged in one period V. Ozolina Econometrics Testing if OLS Assumptions Hold: Example
Graphical test in Excel SUMMARY OUTPUT Regression Statistics Multiple R 0.796991 R Square 0.635195 Adjusted R Square 0.63105 Standard Error 3118.862 Observations 90 ANOVA df SS 1.49E+09 8.56E+08 2.35E+09 MS 1490464776 9727299.3 Coefficients Standard Error 3313.83 1560.228 64.7576 5.231497 t Stat 2.12393914 12.3784059 Regression Residual Total Intercept W_NET_SA 1 88 89 F Significance F 153.2249 5.69E-21 P-value 0.036481 5.69E-21 Lower 95% 213.2041 54.36109 RESIDUAL OUTPUT Observation Predicted IMP_FARMAC_SA 1 13607.51 2 13894.03 3 14214.99 4 14211.98 Residuals -2278.63 -3737.32 -1268.95 -1479.71 -2278.6312 V. -3737.3173 -1268.9512 Ozolina Econometrics Upper 95% Lower 95.0% Upper 95.0% 6414.456 213.2041 6414.456 75.1541 54.36109 75.1541 Testing if OLS Assumptions Hold: Example
Graphical tests in Excel

Do errors group in a circle? V. Ozolina Econometrics Is there a trend? Testing if OLS Assumptions Hold
Cov(ui,uj) = 0 - no autocorrelation
How to test:
Durbin-Watson statistic,
Breusch-Godfrey or serial correlation LM test
H0: assumption holds, Ha: does not hold > 0.05 → 
 

−   < 0.05 → 
    V. Ozolina Econometrics Testing if OLS Assumptions Hold
Durbin-Watson test
H0 : Cov(ui,uj) = 0
H1 : Cov(ui,uj) > 0


The regression model includes an intercept term The X variables are non-stochastic The regression does not contain the lagged value(s) of the dependent variable V. Ozolina Econometrics Testing if OLS Assumptions Hold




Formula of the Durbin-Watson statistics: ∑  −   = ∑   0≤d≤4 Close to 0  Positive autocorrelation Close to 4  Negative autocorrelation Close to 2  No autocorrelation V. Ozolina Econometrics Testing if OLS Assumptions Hold


Formula of the Durbin-Watson statistics: ∑  −   = ∑   0 < d < 4, the closer to 2, the better. If d < 2: If d < dL: H0 is rejected – autocorrelation exists
If d > dU: H0 is accepted – autocorrelation does not exist
If dL < d < dU: we cannot accept or reject H0

If d > 2, then instead of d we use (4 – d) V. Ozolina Econometrics The Durbin-Watson d Statistic Accept H0 or H0* or both Reject H0 Evidence of positive autocorrelation Zone of indecision dL

Reject H0* Zone of indecision dU 2 4 - dU H0: No positive autocorrelation H0*: No negative autocorrelation V. Ozolina Econometrics Evidence of negative autocorrelation 4 - dL 4 The Durbin-Watson d Statistic
Durbin-Watson tables provide:
Lower limit dL
Upper limit dU
Upper and lower limits depend upon:
The number of observations n (6 to 200)
The number of explanatory variables k (up to 20)
Significance levels (1% or 5%) V. Ozolina Econometrics Table of Critical Values V. Ozolina Econometrics Testing if OLS Assumptions Hold: Example
Durbin-Watson test in Excel RESIDUAL OUTPUT 1 2 3 4 5 6 Predicted Observation IMP_FARMAC_SA Residuals Residual (t-1) ([3]-[4])^2 [3]^2 1 13607.51 -2278.63 5192160.1 2 13894.03 -3737.32 -2278.6312 2127765.27 13967541 3 14214.99 -1268.95 -3737.3173 6092831.21 1610237.3 4 14211.98 -1479.71 -1268.9512 44421.3207 2189556.2 88 25362.48 -4292.08 -2148.6509 4594297.91 18421970 89 25333.14 -5845.61 -4292.0823 2413452.82 34171173 90 25352.4 -36.2557 -5845.6114 33748614.2 1314.4731 Sum 1195662701 856002338 DW = (sum([3]-[4])^2) : (sum[3]^2) 1.3967984 V. Ozolina Econometrics Testing if OLS Assumptions Hold: Example
Durbin-Watson test in Excel RESIDUAL OUTPUT 1 2 3 4 5 6 Predicted Observation IMP_FARMAC_SA Residuals Residual (t-1) ([3]-[4])^2 [3]^2 1 13607.51 -2278.63 5192160.1 2 13894.03 -3737.32 -2278.6312 2127765.27 13967541 90 25352.4 -36.2557 -5845.6114 33748614.2 1314.4731 Sum 1195662701 856002338 DW = (sum([3]-[4])^2) : (sum[3]^2) 1.3967984


DW statistics = 1.4 From the table: 90 observations, 2 coefficients, significance level 0.05  dL = 1.635, dU = 1.679 1.39 < 1.635  d < dL  autocorrelation exists V. Ozolina Econometrics Testing if OLS Assumptions Hold: Example Durbin-Watson test in EViews
From the table: 90 observations, Dependent Variable: IMP_FARMAC_SA 2 coefficients, Method: Least Squares Date: 10/22/12 Time: 13:02 significance Sample (adjusted): 2005M01 2012M06 Included observations: 90 after adjustments level 0.05  dL = 1.635, Variable Coefficient Std. Error dU = 1.679 W_NET_SA 64.75760 5.231497
C 1.39 < 1.635  d < dL  autocorrelation exists
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 3313.830 0.635195 0.631050 3118.862 8.56E+08 -850.7633 1.396798 V. Ozolina Econometrics 1560.228 t-Statistic Prob. 12.37841 2.123939 0.0000 0.0365 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 22193.36 5134.666 18.95030 19.00585 153.2249 0.000000 Testing if OLS Assumptions Hold: Example 2 Durbin-Watson test in EViews
From the table: Dependent Variable: UNEMPL Method: Least Squares 18 observations, Date: 04/03/14 Time: 13:11 Sample (adjusted): 1996 2013 2 coefficients, Included observations: 18 after adjustments significance Variable Coefficient Std. Error t-Statistic level 0.05  PCI -0.462323 0.183859 -2.514550 C 9.634877 1.275415 7.554309 dL = 1.158, dU = 1.391 R-squared 0.327225 Mean dependent var
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.275473 2.613244 88.77756 -34.61971 2.097416 (4 – d) = =4 – 2.097 = 1.903
1.903 > 1.39  no autocorrelation
S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) V. Ozolina Econometrics Prob. 0.0259 0.0000 6.913333 3.070102 4.882627 4.977034 6.322960 0.025869 Testing if OLS Assumptions Hold: Example
Serial Correlation LM Test in EViews V. Ozolina Econometrics Testing if OLS Assumptions Hold: Example
Serial Correlation LM Test in Eviews p-value = 0.000
0.000 < 0.05  H0 is rejected  Autocorrelation exists
Breusch-Godfrey Serial Correlation LM Test: F-statistic Obs*R-squared 10.71071 17.94733 Prob. F(2,86) Prob. Chi-Square(2) 0.000070 0.000127 Test Equation: Dependent Variable: RESID Method: Least Squares Date: 10/07/15 Time: 10:11 Sample: 2005M01 2012M06 Included observations: 90 Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-Statistic Prob. W_NET_SA C RESID(-1) RESID(-2) -0.215694 39.54523 0.192388 0.355922 4.735461 1412.212 0.101296 0.103413 -0.045549 0.028002 1.899264 3.441743 0.9638 0.9777 0.0609 0.0009 V. Ozolina Econometrics Testing if OLS Assumptions Hold
Cov(ui,uj) = 0 - no autocorrelation
Reasons for autocorrelation:
Model Specification Error  Under-specification  Wrong functional form
Inertia
Data manipulation  Monthly to quarterly data V. Ozolina Econometrics Testing if OLS Assumptions Hold
Cov(ui,uj) = 0 - no autocorrelation
What to do, if the assumption does not hold:
Find the missing factors,
Check the functional form,
If the form of autocorrelation is known, use GLS (Generalized Least Squares),
If nothing else works, use HAC (Heteroskedasticity consistent coefficient covariance), V. Ozolina Econometrics Testing if OLS Assumptions Hold
Cov(ui,uj) = 0 - no autocorrelation
What to do, if the assumption does not hold:
Specify a dynamic model by introducing the lagged (past) values of y
We have to add lags until the problem is solved V. Ozolina Econometrics Testing if OLS Assumptions Hold: Example

Adding lags in EViews DW close to ideal? Dependent Variable: IMP_FARMAC_SA Method: Least Squares Date: 10/06/15 Time: 12:23 Sample (adjusted): 2005M03 2012M06 Included observations: 88 after adjustments Variable Coefficient Std. Error t-Statistic Prob. W_NET_SA C IMP_FARMAC(-1) IMP_FARMAC(-2) 35.29715 2981.622 0.120906 0.286676 8.830164 1577.625 0.091080 0.091618 3.997337 1.889944 1.327470 3.129050 0.0001 0.0622 0.1879 0.0024 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.659358 0.647193 2902.612 7.08E+08 -824.4760 1.919920 V. Ozolina Econometrics Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 22453.60 4886.751 18.82900 18.94161 54.19785 0.000000 Testing if OLS Assumptions Hold: Example
Adding lags in EViews Breusch-Godfrey Serial Correlation LM Test: LM test:
p-value = 0.79
0.79 > 0.05  autocorrelation problem is solved
F-statistic Obs*R-squared 0.216013 0.461207 Prob. F(2,82) Prob. Chi-Square(2) 0.806182 0.794054 Test Equation: Dependent Variable: RESID Method: Least Squares Date: 10/07/15 Time: 12:27 Sample: 2005M03 2012M06 Included observations: 88 Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-Statistic Prob. W_NET_SA C IMP_FARMAC(-1) IMP_FARMAC(-2) RESID(-1) RESID(-2) 3.111431 223.9373 -0.101750 0.051100 0.141651 -0.038520 11.93342 1639.527 0.184739 0.181037 0.217847 0.207240 0.260733 0.136587 -0.550775 0.282263 0.650234 -0.185873 0.7950 0.8917 0.5833 0.7785 0.5174 0.8530 V. Ozolina Econometrics Testing if OLS Assumptions Hold
A5: ut is normally distributed
Problems:

Several tests are related to the normality assumption, but it usually does not cause problems What to do, if the assumption does not hold: Nothing, if there are a lot of data
If there are separate extreme values, we can use dummies
V. Ozolina Econometrics Evaluation of the Quality of the Equation

Measures of Fit Hypothesis testing for coefficients V. Ozolina Econometrics Measures of Fit
Coefficient of determination R2 – the fraction of the sample variance of Yi explained by Xi
Yi can be split into explained and unexplained part:
0 ≤ R2 ≤ 1
If R2 = 0, Xi explains none of the variation of Yi
If R2 = 1, Xi explains all the variations of Yi:
Standard error SY,X – spread of Yi around the regression line (magnitude of regression error)
The lower the error, the better. V. Ozolina Econometrics Number of crimes per 10 000 residents Components of Total Variation 300 Yi 280 Ŷi = βˆ0 + βˆ1X i Unexplained 260 Explained 240 Total 220 200 180 160 140 120 1000 2000 3000 4000 5000 GDP per capita, Ls V. Ozolina Econometrics 6000 7000 8000 Coefficient of Determination
Ratio of explained to the total sum of squares
Total Sum of Squares
Explained Sum of Squares
Residual Sum of Squares
TSS = ESS + RSS V. Ozolina Econometrics 2 R Coefficient of Determination R2 ...
R2 = (rY,X)2 GDP variations explain 59.2% of variations of crimes
V. Ozolina Econometrics Standard Error SY,X ...




Standard Error of Regression (SER) or Standard Error of Equation (SEE) Standard error SY,X is an estimator of the standard deviation of the regression error ui. SY,X is measured in the units of the dependent variable Yi. It is assumed that , where V. Ozolina Econometrics Measures of Fit: Example
In Excel: Data Analysis Regression SUMMARY OUTPUT Regression Statistics Multiple R 0.796991 R Square 0.635195 Adjusted R Square 0.63105 Standard Error 3118.862 Observations 90 V. Ozolina Econometrics Measures of Fit: Example
In EViews Dependent Variable: IMP_FARMAC_SA Method: Least Squares Date: 10/22/12 Time: 13:02 Sample (adjusted): 2005M01 2012M06 Included observations: 90 after adjustments Variable Coefficient Std. Error t-Statistic Prob. W_NET_SA C 64.75760 3313.830 5.231497 1560.228 12.37841 2.123939 0.0000 0.0365 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.635195 0.631050 3118.862 8.56E+08 -850.7633 1.396798 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) V. Ozolina Econometrics 22193.36 5134.666 18.95030 19.00585 153.2249 0.000000 Hypotheses Tests for Coefficients

The main idea – if the estimated value of the coefficient is valid or is the value considerably different Usually it is applied to evaluate if the chosen factor is statistically significant and thus usable in the equation V. Ozolina Econometrics Two-Sided Hypotheses for β1

Null hypothesis H0: β1 = β1,0 – that true population slope β1 takes on some specific value, β1,0 Two-sided alternative hypothesis H1: β1 ≠ β1,0 – β1 does not equal β1,0 V. Ozolina Econometrics Two-Sided Hypotheses for β1 To test that, we follow 3 steps:
Compute standard error of β1 estimator or standard error
, where
Compute t-statistic
Compute p-value V. Ozolina Econometrics Two-Sided Hypotheses for β1

p-value is the probability of observing a value of at least as different from β1,0 as the estimate actually computed ( ), assuming that the null hypothesis is correct p-value V. Ozolina Econometrics Two-Sided Hypotheses for β1


Null hypothesis is rejected, if the value of p-value is very small (usually smaller than 5%) Null hypothesis is rejected, if calculated value of tstatistic is larger than the critical value of t-statistic Critical value of t-statistic can be found in the Student t Distribution tables, choosing appropriate degrees of freedom (in single regression = n-2) and probability or p-value (usually 0.05 or 0.01)
In Excel we can use function TINV or T.INV.2T V. Ozolina Econometrics Two-Sided Hypotheses for β1: Example
Intercept W_NET_SA In Excel: Data Analysis Regression Coefficients 3313.83 64.7576 Intercept W_NET_SA Standard Error 1560.228 5.231497 t Stat 2.12393914 12.3784059 P-value 0.036481 5.69E-21 Lower 95% 213.2041 54.36109 Upper 95% 6414.456 75.1541 Coefficients Standard Error t Stat 3313.83 1560.228 2.12393914 64.7576 5.231497 12.3784059 V. Ozolina Econometrics Lower 95.0% 213.2041 54.36109 Upper 95.0% 6414.456 75.1541 P-value 0.036481 0.000000 Two-Sided Hypotheses for β1: Example
In EViews Dependent Variable: IMP_FARMAC_SA Method: Least Squares Date: 10/22/12 Time: 13:02 Sample (adjusted): 2005M01 2012M06 Included observations: 90 after adjustments Variable Coefficient Std. Error t-Statistic Prob. W_NET_SA C 64.75760 3313.830 5.231497 1560.228 12.37841 2.123939 0.0000 0.0365 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.635195 0.631050 3118.862 8.56E+08 -850.7633 1.396798 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) V. Ozolina Econometrics 22193.36 5134.666 18.95030 19.00585 153.2249 0.000000 Two-Sided Hypotheses Concerning β1...



Example – import of pharmaceutical products depending on wages y = 64.758x + 3313.8 + u Standard error of coefficient β1 = 5.23  t-stat = (64.758-0)/5.23 = 12.4 Critical value of t-statistic:



Degrees of freedom = n – 2 = 90 – 2 = 88 probability (also α) = 0.05 Critical value – 1.987 12.4 > 1.987  reject null hypothesis

(p-value < 0,0000) Coefficient β1 is statistically significant V. Ozolina Econometrics Two-Sided Hypotheses Concerning β1
Reporting regression equations: Imp_farm_sa = 64.758w_net_sa + 3313.8 SE (5.23) (1560.2) R2 = 0.64; SER = 3118.9 or
Imp_farm_sa = 64.758w_net_sa + 3313.8 t-stat (12.4) (2.1) R2 = 0.64 [01/2005 – 06/2013]
V. Ozolina Econometrics One-Sided Hypotheses for β1

Test, whether the value can be >0 or <0 (depending on the direction of the factor) Rarely used in practice V. Ozolina Econometrics Hypotheses for the Intecrept β0
Procedures are the same as when testing β1, however, it is done only if you have a specific null hypothesis in mind. V. Ozolina Econometrics Confidence Intervals for Regression Coefficients

It is possible to use the OLS estimator and its standard error to construct a confidence intervals for regression coefficients with a large degree of confidence 95% confidence interval for β1 means that:
It is the set of values of β1 that cannot be rejected using a two-sided hypothesis test with a 5% significance level
It is an interval that has a 95% probability of containing the true value of β1 V. Ozolina Econometrics Confidence Intervals for Regression Coefficients


! ∓ 1.987'(( ! ) y = 64.758x + 3313.8 + u 95% confidence interval for slope is
[64.758 – 1.987*5.23; 64.758 + 1.987*5.23]
[54.36; 75.15] V. Ozolina Econometrics Confidence Intervals for Regression Coefficients: Example
Intercept W_NET_SA In Excel: Data Analysis Regression Coefficients 3313.83 64.7576 Standard Error 1560.228 5.231497 t Stat 2.12393914 12.3784059 P-value 0.036481 5.69E-21 Lower 95% 213.2041 54.36109 Lower 95% Upper 95% 213.2041 6414.456 54.36109 75.1541 V. Ozolina Econometrics Upper 95% 6414.456 75.1541 Lower 95.0% 213.2041 54.36109 Upper 95.0% 6414.456 75.1541