Autoregressive Conditional Heteroskedasticity

Financial time-series such as foreign exchange rates, inflation rates and stock prices may exhibit some volatility which varies over time. In the case of inflation or foreign exchange rates this could be due to changes in the Federal Reserve’s policies. In the case of stock prices this could be due to rumors about a certain company’s merger or takeover. This suggests that the variance of these time-series may be heteroskedastic. Engle (1982) modeled this heteroskedasticity by relating the conditional variance of the disturbance term at time t to the size of the squared disturbance terms in the recent past. A simple Autoregressive Conditionally Heteroskedastic
(ARCH) model is given by

= E(ut/( t) = Yo + Y iut-i + •• + Yp^t-v (14-22)

where Zt denotes the information set upon which the variance of ut is to be conditioned. This typically includes all the information available prior to period t. In (14.22), the variance of ut conditional on the information prior to period t is an autoregressive function of order p in squared lagged values of ut. This is called an ARCH(p) process. Since (14.22) is a variance, this means that all the Y^s for i = 0,1,-,p have to be non-negative. Engle (1982) showed that a simple test for homoskedasticity, i. e., Ho; yi = Y2 = •• = Yv = 0, can be based upon an ordinary F-test which regresses the squared OLS residuals (e2) on their lagged values (e^^-^e2^) and a constant. The F-statistic tests the joint significance of the regressors and is reported by most regression packages. Alternatively, one can compute T times the centered R2 of this regression and this is distributed as xp under the null hypothesis Ho. This test resembles the usual homoskedasticity tests studied in Chapter 5 except that the squared OLS residuals are regressed upon their lagged values rather than some explanatory variables.

The simple ARCH(1) process

at = Yo + Yiut-1 (14.23)

can be generated as follows: ut = [yo + Y 1ut-i1/‘2€t where et ~ IID(0,1). Note that the sim­plifying variance of unity for et can be achieved by rescaling the parameters yo and y1. In this case, the conditional mean of ut is given by

E(ut/Z t) = [Y o + Y iu2-i}1/2 E (Y/Z t) = 0

since ut 1 is known at time t. Similarly, the conditional variance can be easily obtained from

E(u2/Z t) = [Y o + Y 1u2-1}E (H2/C t) = Yo + Y 1u2-1

since E(e‘2) = 1. Also, the conditional covariances can be easily shown to be zero since

E(utut-s/Z t )= ut-s E(ut/Z t) = 0 for s = 1, 2,^,t

The unconditional mean can be obtained by taking repeated conditional expectations period by period until we reach the initial period, see the Appendix to Chapter 2. For example, taking the conditional expectation of E(ut/Zt) based on information prior to period t — 1, we get

E[E(ut/Zt)/Zt-1 = E (0/Zt-1) = 0

It is clear that all prior conditional expectations of zero will be zero so that E(ut) = 0. Similarly, taking the conditional expectations of E(u2/Zt) based on information prior to period t — 1, we

Подпись: E[E(U2/Z t)/Z t-1} = Yo + Y1E K2-1/Z t-1 Подпись: Yo + Y 1(Yo + Y 1ut-2) = Y o(1 + Y 1) + Y1 ut-2

get

Подпись: (14.24)By taking repeated conditional expectations one period at a time we finally get E(ut) = Yo(1 + Y1 + Y1 + •• + Y 1-1) + Y1uo

As t ^ x, the unconditional variance of ut is given by a2 = var(ut) = 70/(1 — Y1) for |y11 < 1 and Yo > 0. Therefore, the ARCH(1) process is homoskedastic.

ARCH models can be estimated using feasible GLS or maximum likelihood methods. Alterna­tively, one can use a double-length regression procedure suggested by Davidson and MacKinnon (1993) to obtain (i) one-step efficient estimates starting from OLS estimates or (ii) the max­imum likelihood estimates. Here we focus on the feasible GLS procedure suggested by Engle (1982). For the regression model

y = Хв + u (14.25)

where y is T x 1 and X is T x k. First, obtain the OLS estimates вOLS and the OLS residuals e. Second, perform the following regression: ef = ao + a+e‘2-1 + residuals. This yields a test for homoskedasticity. Third, compute af = ao + a1 e2-1 and regress [(e2/at) — 1] on (1/at) and

Подпись: rt Подпись: 1 —+ 2 at image621 Подпись: 21 1/2 Подпись: (14.26)

(ef-1 /at). Call the regression estimates da. One updates a’ = (ao, a1) by computing a = a + da. Fourth, recompute?2 using the updated a from step 3, and form the set of regressors xtjrt for j = 1,…,k, where

Finally, regress (etst/rt) where

st = 1 — 71 (X —

at at+1 at+1

on xtjrt for j = 1,…,k and obtain the least squares coefficients dy. Update the estimate of в by computing [3 = eOLS + dp. This procedure can run into problems if the are not all positive, see Judge et al. (1985) and Engle (1982) for details.

The ARCH model has been generalized by Bollerslev (1986). The Generalized ARCH (GARCH (p, q)) model can be written as

Подпись: (14.27)a2 = Yo + E i=1 Yiu-i + E q=1 Sj a2t-j

In this case, the conditional variance of ut depends upon q of its lagged values as well as p squared lagged values of ut. The simple GARCH (1,1) model is given by

Подпись:2 і 2 і c 2

at = Yo + Y1ut-1 + S1at-1

An LM test for GARCH (p, q) turns out to be equivalent to testing ARCH (p + q). This simply regresses squared OLS residuals on (p + q) of its squared lagged values. The test statistic is T times the uncentered R2 and is asymptotically distributed as Xp+q under the null of homoskedasticity.

In conclusion, a lot of basic concepts have been introduced in this chapter and we barely scratched the surface. Hopefully, this will motivate the reader to take the next econometrics time series course.

Table 14.3 GARCH (1,1) model

Dependent Variable: CONSUMP

Method: ML – ARCH (Marquardt) – Normal distribution

Sample: 1959 2007 Included observations:

49

Convergence achieved after 19 iterations Presample variance: backcast (parameter = 0.7)

GARCH = C(3) + C(4)*RES! D(-1)"2 + C(5)*GARCH(-1)

Coefficient

Std. Error

z-Statistic

Prob.

C

-1435.888

226.3933

-6.342449

0.0000

Y

0.986813

0.011923

82.76452

0.0000

Variance Equation

C

118728.2

87402.35

1.358410

0.1743

RESID(-1)~ 2

1.068561

0.326091

3.276885

0.0010

GARCH(-1)

-0.380949

0.187656

-2.030036

0.0424

R-squared

0.993542

Mean dependent var

16749.10

Adjusted R-squared

0.992955

S. D. dependent var

5447.060

S. E. of regression

457.1938

Akaike info criterion

14.86890

Sum squared resid

9197153.

Schwarz criterion

15.06194

Log likelihood

-359.2880

Hannah-Quinn criter.

14.94214

F-statistic

1692.353

Durbin-Watson stat

0.178409

Prob(F-statistic)

0.000000

Note

1. Granger causality has been developed by Granger (1969). For

another definition of causality,

see Sims (1972).

Also, Chamberlain (1982)

for a discussion on

when these

two definitions are

equivalent.

Problems

1. For the AR(1) model

yt = PVt-1 + et t = 1, 2,…,T; with p < 1 and et ~ IIN(0,ct^ )

(a) Show that if yo ~ N(0,a2e/1 — p2), then E(yt) = 0 for all t and var(yt) = oj(1 — p2) so that the mean and variance are independent of t. Note that if p =1 then var(yt) is to. If p > 1 then var(yt) is negative!

(b) Show that cov(yt, yt-s) = psa2 which is only dependent on s, the distance between the two time periods. Conclude from parts (a) and (b) that this AR(1) model is weakly stationary.

(c) Generate the above AR(1) series for T = 250, a2 = 0.25 and various values of p = ±0.9, ±0.8, ±0.5, ±0.3 and ±0.1. Plot the AR(1) series and the autocorrelation function ps versus s.

2. For the MA(1) model

yt = et + Oe—1 t = 1, 2,…,T; with et ~ IIN(0, a2e)

(a) Show that E(yt) = 0 and var(yt) = 1 + в2) so that the mean and variance are independent of t.

(b) Show that cov(yt, yt-1) = ва2 and cov(yt, yt-s) = 0 for s > 1 which is only dependent on s, the distance between the two time periods. Conclude from parts (a) and (b) that this MA(1) model is weakly stationary.

(c) Generate the above MA(1) series for T = 250, a2 = 0.25 and various values of в = 0.9, 0.8, 0.5, 0.3 and 0.1. Plot the MA(1) series and the autocorrelation function versus s.

3. Using the consumption-personal disposable income data for the U. S. used in this chapter:

(a) Compute the sample autocorrelation function for personal disposable income (Yt). Plot the sample correlogram. Repeat for the first-differenced series (AYt). Compute the Ljung-Box Qlb statistic, test that Ho; ps = 0 for s = 1,…, 20.

(b) Run the Augmented Dickey-Fuller test for the existence of a unit root in personal disposable income (Yt).

(c) Define Yt = AYt and run AYt on Yt-1 and a constant and trend. Test that the first-differenced series of personal disposable income is stationary. What do you conclude? Is Yt an I(1) process?

(d) Replicate the regression in (14.21) and verify the Engle-Granger (1987) test for cointegration.

(e) Replicate the GARCH(1,1) model given in Table 14.3.

(f) Repeat parts (a) through (e) using logC and logY. Are there any changes in the above results?

4. (a) Generate T = 25 observations on xt and yt as independent random walks with IIN(0,1)

disturbances. Run the regression yt = a + /3xt + ut and test the null hypothesis Ho; в = 0 using the usual t-statistic at the 1%, 5% and 10% levels. Repeat this experiment 1000 times and report the frequency of rejections at each significance level. What do you conclude?

(b) Repeat part (a) for T = 100 and T = 500.

(c) Repeat parts (a) and (b) generating xt and yt as independent random walks with drift as described in (14.11), using IIN(0,1) disturbances. Let 7 = 0.2 for both series.

(d) Repeat parts (a) and (b) generating xt and yt as independent trend stationary series as described in (14.10), using IIN(0,1) disturbances. Let a =1 and в = 0.04 for both series.

(e) Report the frequency distributions of the R2 statistics obtained in parts (a) through (d) for each sample size and method of generating the time-series. What do you conclude? Hint: See the Monte Carlo experiments in Granger and Newbold (1974), Davidson and MacKinnon (1993) and Banerjee, Dolado, Galbraith and Hendry (1993).

5. For the Money Supply, GNP and interest rate series data for the U. S. given on the Springer web site as MACRO. ASC, fit a VAR three equation model using:

(a) Two lags on each variable.

(b) Three lags on each variable.

(c) Compute the Likelihood Ratio test for part (a) versus part (b).

(d) For the two-equation VAR of Money Supply and interest rate with three lags on each variable, test that the interest rate does not Granger cause the money supply?

(e) How sensitive are the tests in part (d) if we had used only two lags on each variable.

6. For the simple Deterministic Time Trend Model Vt = a + fit + Ut t = 1, ..,T

where ut ~ IIN(0,v2).

(a) Show that

Подпись:Подпись: (X 'X )X 'uaOLS — a

Pols — P

where the t-th observation of X, the matrix of regressors, is [1,t].

(b) Use the results that ^t=i t = T(T + 1)/2 and t=i t2 = T(T + 1)(2T + 1)/6 to show that

plim (X’X/T) as T is not a positive definite matrix.

(c) Use the fact that

Подпись: T (aOLS — a) V Ty/T(POLS — P)

image629
Подпись: where A

A(X ‘X )-1AA-1 (X’u) = (A-1(X ‘X )A-1)-1A-1(X’u)

(d) Подпись:Show that z1 = f=1 ut/VT is N(0,v2) and z2

1)/6T2) with cov(z1,z2) = (T + 1)<t2/2T, so that

Подпись: / ( 1 T + 1  0, V2 2T T +1 (T +1)(2T +1)  V 2T 6T2 ) ) z1

Подпись: Conclude that as T ^<x>, the asymptotic distribution of Подпись: z1 z2 Подпись: is N(0, V2Q).

z2

(e) Using the results in parts (c) and (d), conclude that the asymptotic distribution of

rVT^a°LS ^ is N(0,a2Q-1). Since (3OLS has the factor TVT rather than the usual

T T(PoLS – P) )

VT, it is said to be superconsistent. This means that not only does (POLS — P) converge to zero in probability limits, but so does T(POLS — P). Note that the normality assumption is not needed for this result. Using the central limit theorem, all that is needed is that ut is White noise with finite fourth moments, see Sims, Stock and Watson (1990) or Hamilton (1994).

7. Test of Hypothesis with a Deterministic Time Trend Model. This is based on Hamilton (1994). In problem 6, we showed that aOLS and POLS converged at different rates, %/T and TVT respectively. Despite this fact, the usual least squares t and F-statistics are asymptotically valid even when the ut’s are not Normally distributed.

(a) Show that s2 = ^’T=1(yt — aOLS — Polst)2/(T — 2) has plim s2 = a2.

(b) In order to test Ho; a = ao, the usual least squares package computes

ta = (Sols — ao)/[s2(1,0)(X’X)-1(1, 0)’]1/2

where (X’X) is given in problem 6. Multiply the numerator and denominator by %/T and use the results of part (c) of problem 6 to show that this t-statistic has the same asymptotic distribution as t*a = %/T(aOLS — ao)/aZqu where q11 is the (1, 1) element of Q-1 defined in problem 6. t*a has an asymptotic N(0,1) distribution using the results of part (e) in problem 6.

(c) Similarly, to test Ho; в = во, the usual least squares package computes

te = (Pols — e)/[s2(0,1)(X ‘X )-1 (0,1)’]1/2.

Multiply the numerator and denominator by T/T and use the results of part (c) of problem 6 to show that this t-statistic has the same asymptotic distribution as t*p = T/T(POLS — P)/a/q22 where q22 is the (2, 2) element of Q-1 defined in problem 6. t*g has an asymptotic N(0, 1) distribution using the results of part (e) in problem 6.

8. A Random Walk Model. This is based on Fuller (1976) and Hamilton (1994). Consider the following random walk model

yt = yt-1 + ut t = 0,1,…,T where ut ~ IIN(0, a2) and yo = 0.

(a) Show that yt can be written as yt = u1 + u2 + .. + ut with E(yt) = 0 and var(yt) = ta2 so that yt ~ N(0,ta2).

(b) Square the random walk equation yj = (yt-1 + ut)2 and solve for yt-1ut. Sum this over t = 1, 2,.. .,T and show that

St=1 yt-1ut = (yT/2) — ^21=1 ut/2

Divide by Ta2 and show that ^T=1 yt-1ut/Ta2 is asymptotically distributed as (y2 — 1)/2. Hint: Use the fact that yT ~ N(0,Ta2).

(c) Using the fact that yt-1 ~ N(0, (t — 1)a2) show that E (Y^t=1 y‘t-1j = a2T(T — 1)/2. Hint: Use the expression for ^T=11 in problem 6.

(d) Suppose we had estimated an AR(1) model rather than a random walk, i. e., yt = pyt-1 + ut when the true p = 1. The OLS estimate is

X^T X^T 2 і X^T X^T 2

p = t=1 yt-1 rytl t=1 yt-1 = p + t=1 yt-1 ut/ t=1 yt-1

Show that

t=1 yt-1ut/Ta

plim T(p — p) = plim <tTL 2 2 2 = 0

t=1 y – 1/T 2 a2

Note that the numerator was considered in part (b), while the denominator was considered in part (c). One can see that the asymptotic distribution of p when p = 1 is a ratio of (x2 — 1)/2 random variable to a non-standard distribution in the denominator which is beyond the scope of this book, see Hamilton (1994) or Fuller (1976) for further details. The object of this exercise is to show that if p = 1, VT(s — p) is no longer normal as in the standard stationary least squares regression with p < 1. Also, to show that for the nonstationary (random walk) model, p converges at a faster rate (T) than for the stationary case (%/T). From part (c) it is clear that one has to divide the denominator of p by T2 rather than T to get a convergent distribution.

9. Consider the cointegration example given in (14.13) and (14.14).

(a) Verify equations (14.15)-(14.20).

(b) Show that the OLS estimator of в obtained by regressing Ct on Yt is superconsistent, i. e., show that plim T(Pols — в) ^ 0 as T ^x>.

References

This chapter draws on the material in Davidson and MacKinnon (1993), Maddala (1992), Hamilton

(1994), Banerjee et al. (1993) and Gujarati (1995). Advanced readings include Fuller (1976) and Hamilton

(1994). Easier readings include Mills (1990) and Enders (1995).

Banerjee, A., J. J. Dolado, J. W. Galbraith and D. F. Hendry (1993), Co-Integration, Error-Correction, and The Econometric Analysis of Non-stationary Data (Oxford University Press: Oxford).

Bierens, H. J. (2001), “ Unit Roots,” Chapter 29 in B. H. Baltagi (ed.) A Companion to Theoretical Econometrics (Blackwell: Massachusetts).

Bierens, H. J. and S. Guo (1993), “Testing for Stationarity and Trend Stationarity Against the Unit Root Hypothesis,” Econometric Reviews, 12: 1-32.

Bollerslev, T. (1986), “Generalized Autoregressive Heteroskedasticity,” Journal of Econometrics, 31: 307-327.

Box, G. E.P. and G. M. Jenkins (1970), Time Series Analysis, Forecasting and Control (Holden Day: San Francisco).

Box, G. E.P. and D. A. Pierce (1970), “The Distribution of Residual Autocorrelations in Auto-regressive – Integrated Moving Average Time Series Models,” Journal of American Statistical Association, 65: 1509-1526.

Chamberlain, G. (1982), “The General Equivalence of Granger and Sims Causality,” Econometrica, 50: 569-582.

Davidson, R. and J. G. MacKinnon (1993), Estimation and Inference in Econometrics (Oxford University Press: Oxford).

Dickey, D. A. and W. A. Fuller (1979), “Distribution of the Estimators for Autoregressive Time Series with A Unit Root,” Journal of the American Statistical Association, 74: 427-431.

Dolado, J. J., J. Gonzalo and F. Marmol (2001), “Cointegration,” Chapter 30 in B. H. Baltagi (ed.) A Companion to Theoretical Econometrics (Blackwell: Massachusetts).

Durlauf, S. N. and P. C.B. Phillips (1988), “Trends versus Random Walks in Time Series Analysis,” Econometrica, 56: 1333-1354.

Enders, W. (1995), Applied Econometric Time Series (Wiley: New York).

Engle, R. F. (1982), “Autogressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation,” Econometrica, 50: 987-1007.

Engle, R. F. and C. W.J. Granger (1987), “Co-Integration and Error Correction: Representation, Estima­tion and Testing,” Econometrica, 55: 251-276.

Fuller, W. A. (1976), Introduction to Statistical Time Series (John Wiley and Sons: New York).

Geweke, J., R. Meese and W. Dent (1983), “Comparing Alternative Tests of Causality in Temporal Systems: Analytic Results and Experimental Evidence,” Journal of Econometrics, 21: 161-194.

Ghysels, E. and P. Perron (1993), “The Effect of Seasonal Adjustment Filters on Tests for a Unit Root,” Journal of Econometrics, 55: 57-98.

Godfrey, L. G. (1979), “Testing the Adequacy of a Time Series Model,” Biometrika, 66: 67-72.

Granger, C. W.J. (1969), “Investigating Causal Relations by Econometric Models and Cross-Spectral Methods,” Econometrica, 37: 424-438.

Granger, C. W.J. (2001), “Spurious Regressions in Econometrics,” Chapter 26 in B. H. Baltagi (ed.) A Companion to Theoretical Econometrics (Blackwell: Massachusetts).

Granger, C. W.J., M. L. King and H. White (1995), “Comments on Testing Economic Theories and the Use of Model Selection Criteria,” Journal of Econometrics, 67: 173-187.

Granger, C. W.J. and P. Newbold (1974), “Spurious Regressions in Econometrics,” Journal of Econo­metrics, 2: 111-120.

Gujarati, D. N. (1995), Basic Econometrics (McGraw Hill: New York).

Hamilton, J. D. (1994), Time Series Analysis (Princeton University Press: Princeton, New Jersey).

Johansen, S. (1988), “Statistical Analysis of Cointegrating Vectors,” Journal of Economic Dynamics and Control, 12: 231-254.

Judge, G. G., R. C. Hill, W. E. Griffiths, H. Liitkepohl and T. C. Lee (1985), The Theory and Practice of Econometrics (John Wiley and Sons: New York).

Kwaitowski, D., P. C.B. Phillips, P. Schmidt and Y. Shin (1992), “Testing the Null Hypothesis of Sta – tionarity Against the Alternative of a Unit Root,” Journal of Econometrics, 54: 159-178.

Leybourne, S. J. and B. P.M. McCabe (1994), “A Consistent Test for a Unit Root,” Journal of Business and Economic Statistics, 12: 157-166.

Litterman, R. B. (1986), “Forecasting with Bayesian Vector Autoregressions-Five Years of Experience,” Journal of Business and Economic Statistics, 4: 25-38.

Ljung, G. M. and G. E.P. Box (1978), “On a Measure of Lack of Fit in Time-Series Models,” Biometrika, 65: 297-303.

Liitkepohl, H. (2001), “Vector Autoregressions,” Chapter 32 in B. H. Baltagi (ed.) A Companion to Theoretical Econometrics (Blackwell: Massachusetts).

MacKinnon, J. G. (1991), ’’Critical Values for Cointegration Tests,” Ch. 13 in Long-Run Economic Re­lationships: Readings in Cointegration, eds. R. F. Engle and C. W.J. Granger (Oxford University Press: Oxford ).

Maddala, G. S. (1992), Introduction to Econometrics (Macmillan: New York).

Mills, T. C. (1990), Time Series Techniques for Economists (Cambridge University Press: Cambridge).

Nelson, C. R. and C. I. Plosser (1982), “Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implications,” Journal of Monetary Economics, 10: 139-162.

Ng, S. and P. Perron (1995), “Unit Root Tests in ARMA Models With Data-Dependent Methods for the Selection of the Truncation Lag,” Journal of the American Statistical Association, 90: 268-281.

Perron, P. (1989), “The Great Cash, The Oil Price Shock, and the Unit Root Hypothesis,” Econometrica, 57: 1361-1401.

Phillips, P. C.B. (1986), “Understanding Spurious Regressions in Econometrics,” Journal of Economet­rics, 33: 311-340.

Phillips, P. C.B. and P. Perron (1988), “Testing for A Unit Root in Time Series Regression,” Biometrika, 75: 335-346.

Plosser, C. I. and G. W. Shwert (1978), “Money, Income and Sunspots: Measuring Economic Relationships and the Effects of Differencing,” Journal of Monetary Economics, 4: 637-660.

Sims, C. A. (1972), “Money, Income and Causality,” American Economic Review, 62: 540-552.

Sims, C. A. (1980), “Macroeconomics and Reality,” Econometrica, 48: 1-48.

Sims, C. A., J. H. Stock and M. W. Watson (1990), “Inference in Linear Time Series Models with Some Unit Roots,” Econometrica, 58: 113-144.

Stock, J. H. and M. W. Watson (1988), “Variable Trends in Economic Time Series,” Journal of Economic Perspectives, 2: 147-174.

Подпись: Appendix
image637

0 z

Ф(1.65) = pr[z < 1.65] = 0.9505

Table A Area under the Standard Normal Distribution

z

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.0

0.5000

0.5040

0.5080

0.5120

0.5160

0.5199

0.5239

0.5279

0.5319

0.5359

0.1

0.5398

0.5438

0.5478

0.5517

0.5557

0.5596

0.5636

0.5675

0.5714

0.5753

0.2

0.5793

0.5832

0.5871

0.5910

0.5948

0.5987

0.6026

0.6064

0.6103

0.6141

0.3

0.6179

0.6217

0.6255

0.6293

0.6331

0.6368

0.6406

0.6443

0.6480

0.6517

0.4

0.6554

0.6591

0.6628

0.6664

0.6700

0.6736

0.6772

0.6808

0.6844

0.6879

0.5

0.6915

0.6950

0.6985

0.7019

0.7054

0.7088

0.7123

0.7157

0.7190

0.7224

0.6

0.7257

0.7291

0.7324

0.7357

0.7389

0.7422

0.7454

0.7486

0.7517

0.7549

0.7

0.7580

0.7611

0.7642

0.7673

0.7704

0.7734

0.7764

0.7794

0.7823

0.7852

0.8

0.7881

0.7910

0.7939

0.7967

0.7995

0.8023

0.8051

0.8078

0.8106

0.8133

0.9

0.8159

0.8186

0.8212

0.8238

0.8264

0.8289

0.8315

0.8340

0.8365

0.8389

1.0

0.8413

0.8438

0.8461

0.8485

0.8508

0.8531

0.8554

0.8577

0.8599

0.8621

1.1

0.8643

0.8665

0.8686

0.8708

0.8729

0.8749

0.8770

0.8790

0.8810

0.8830

1.2

0.8849

0.8869

0.8888

0.8907

0.8925

0.8944

0.8962

0.8980

0.8997

0.9015

1.3

0.9032

0.9049

0.9066

0.9082

0.9099

0.9115

0.9131

0.9147

0.9162

0.9177

1.4

0.9192

0.9207

0.9222

0.9236

0.9251

0.9265

0.9279

0.9292

0.9306

0.9319

1.5

0.9332

0.9345

0.9357

0.9370

0.9382

0.9394

0.9406

0.9418

0.9429

0.9441

1.6

0.9452

0.9463

0.9474

0.9484

0.9495

0.9505

0.9515

0.9525

0.9535

0.9545

1.7

0.9554

0.9564

0.9573

0.9582

0.9591

0.9599

0.9608

0.9616

0.9625

0.9633

1.8

0.9641

0.9649

0.9656

0.9664

0.9671

0.9678

0.9686

0.9693

0.9699

0.9706

1.9

0.9713

0.9719

0.9726

0.9732

0.9738

0.9744

0.9750

0.9756

0.9761

0.9767

2.0

0.9772

0.9778

0.9783

0.9788

0.9793

0.9798

0.9803

0.9808

0.9812

0.9817

2.1

0.9821

0.9826

0.9830

0.9834

0.9838

0.9842

0.9846

0.9850

0.9854

0.9857

2.2

0.9861

0.9864

0.9868

0.9871

0.9875

0.9878

0.9881

0.9884

0.9887

0.9890

2.3

0.9893

0.9896

0.9898

0.9901

0.9904

0.9906

0.9909

0.9911

0.9913

0.9916

2.4

0.9918

0.9920

0.9922

0.9925

0.9927

0.9929

0.9931

0.9932

0.9934

0.9936

2.5

0.9938

0.9940

0.9941

0.9943

0.9945

0.9946

0.9948

0.9949

0.9951

0.9952

2.6

0.9953

0.9955

0.9956

0.9957

0.9959

0.9960

0.9961

0.9962

0.9963

0.9964

2.7

0.9965

0.9966

0.9967

0.9968

0.9969

0.9970

0.9971

0.9972

0.9973

0.9974

2.8

0.9974

0.9975

0.9976

0.9977

0.9977

0.9978

0.9979

0.9979

0.9980

0.9981

2.9

0.9981

0.9982

0.9982

0.9983

0.9984

0.9984

0.9985

0.9985

0.9986

0.9986

3.0

0.9987

0.9987

0.9987

0.9988

0.9988

0.9989

0.9989

0.9989

0.9990

0.9990

Source: The SAS® function PROBNORM was used to generate this table.

B. H. Baltagi, Econometrics, Springer Texts in Business and Economics, DOI 10.1007/978-3-642-20059-5, © Springer-Verlag Berlin Heidelberg 2011

image638

Pr[t8 >ta = 2.306] = 0.025

Table B Right-Tail Critical Values for the t-Distribution

DF

a=0.1

a=0.05

a=0.025

a=0.01

a=0.005

1

3.0777

6.3138

12.7062

31.8205

63.6567

2

1.8856

2.9200

4.3027

6.9646

9.9248

3

1.6377

2.3534

3.1824

4.5407

5.8409

4

1.5332

2.1318

2.7764

3.7469

4.6041

5

1.4759

2.0150

2.5706

3.3649

4.0321

6

1.4398

1.9432

2.4469

3.1427

3.7074

7

1.4149

1.8946

2.3646

2.9980

3.4995

8

1.3968

1.8595

2.3060

2.8965

3.3554

9

1.3830

1.8331

2.2622

2.8214

3.2498

10

1.3722

1.8125

2.2281

2.7638

3.1693

11

1.3634

1.7959

2.2010

2.7181

3.1058

12

1.3562

1.7823

2.1788

2.6810

3.0545

13

1.3502

1.7709

2.1604

2.6503

3.0123

14

1.3450

1.7613

2.1448

2.6245

2.9768

15

1.3406

1.7531

2.1314

2.6025

2.9467

16

1.3368

1.7459

2.1199

2.5835

2.9208

17

1.3334

1.7396

2.1098

2.5669

2.8982

18

1.3304

1.7341

2.1009

2.5524

2.8784

19

1.3277

1.7291

2.0930

2.5395

2.8609

20

1.3253

1.7247

2.0860

2.5280

2.8453

21

1.3232

1.7207

2.0796

2.5176

2.8314

22

1.3212

1.7171

2.0739

2.5083

2.8188

23

1.3195

1.7139

2.0687

2.4999

2.8073

24

1.3178

1.7109

2.0639

2.4922

2.7969

25

1.3163

1.7081

2.0595

2.4851

2.7874

26

1.3150

1.7056

2.0555

2.4786

2.7787

27

1.3137

1.7033

2.0518

2.4727

2.7707

28

1.3125

1.7011

2.0484

2.4671

2.7633

29

1.3114

1.6991

2.0452

2.4620

2.7564

30

1.3104

1.6973

2.0423

2.4573

2.7500

31

1.3095

1.6955

2.0395

2.4528

2.7440

32

1.3086

1.6939

2.0369

2.4487

2.7385

33

1.3077

1.6924

2.0345

2.4448

2.7333

34

1.3070

1.6909

2.0322

2.4411

2.7284

35

1.3062

1.6896

2.0301

2.4377

2.7238

36

1.3055

1.6883

2.0281

2.4345

2.7195

37

1.3049

1.6871

2.0262

2.4314

2.7154

38

1.3042

1.6860

2.0244

2.4286

2.7116

39

1.3036

1.6849

2.0227

2.4258

2.7079

40

1.3031

1.6839

2.0211

2.4233

2.7045

V-2/v

1

2

3

4

5

6

7

8

9

10

12

15

20

25

30

40

1

161.448

199.500

215.707

224.583

230.162

233.986

236.768

238.883

240.543

241.882

243.906

245.950

248.013

249.260

250.095

251.143

2

18.513

19.000

19.164

19.247

19.296

19.330

19.353

19.371

19.385

19.396

19.413

19.429

19.446

19.456

19.462

19.471

3

10.128

9.552

9.277

9.117

9.013

8.941

8.887

8.845

8.812

8.786

8.745

8.703

8.660

8.634

8.617

8.594

4

7.709

6.944

6.591

6.388

6.256

6.163

6.094

6.041

5.999

5.964

5.912

5.858

5.803

5.769

5.746

5.717

5

6.608

5.786

5.409

5.192

5.050

4.950

4.876

4.818

4.772

4.735

4.678

4.619

4.558

4.521

4.496

4.464

6

5.987

5.143

4.757

4.534

4.387

4.284

4.207

4.147

4.099

4.060

4.000

3.938

3.874

3.835

3.808

3.774

7

5.591

4.737

4.347

4.120

3.972

3.866

3.787

3.726

3.677

3.637

3.575

3.511

3.445

3.404

3.376

3.340

8

5.318

4.459

4.066

3.838

3.687

3.581

3.500

3.438

3.388

3.347

3.284

3.218

3.150

3.108

3.079

3.043

9

5.117

4.256

3.863

3.633

3.482

3.374

3.293

3.230

3.179

3.137

3.073

3.006

2.936

2.893

2.864

2.826

10

4.965

4.103

3.708

3.478

3.326

3.217

3.135

3.072

3.020

2.978

2.913

2.845

2.774

2.730

2.700

2.661

11

4.844

3.982

3.587

3.357

3.204

3.095

3.012

2.948

2.896

2.854

2.788

2.719

2.646

2.601

2.570

2.531

12

4.747

3.885

3.490

3.259

3.106

2.996

2.913

2.849

2.796

2.753

2.687

2.617

2.544

2.498

2.466

2.426

13

4.667

3.806

3.411

3.179

3.025

2.915

2.832

2.767

2.714

2.671

2.604

2.533

2.459

2.412

2.380

2.339

14

4.600

3.739

3.344

3.112

2.958

2.848

2.764

2.699

2.646

2.602

2.534

2.463

2.388

2.341

2.308

2.266

15

4.543

3.682

3.287

3.056

2.901

2.790

2.707

2.641

2.588

2.544

2.475

2.403

2.328

2.280

2.247

2.204

16

4.494

3.634

3.239

3.007

2.852

2.741

2.657

2.591

2.538

2.494

2.425

2.352

2.276

2.227

2.194

2.151

17

4.451

3.592

3.197

2.965

2.810

2.699

2.614

2.548

2.494

2.450

2.381

2.308

2.230

2.181

2.148

2.104

18

4.414

3.555

3.160

2.928

2.773

2.661

2.577

2.510

2.456

2.412

2.342

2.269

2.191

2.141

2.107

2.063

19

4.381

3.522

3.127

2.895

2.740

2.628

2.544

2.477

2.423

2.378

2.308

2.234

2.155

2.106

2.071

2.026

20

4.351

3.493

3.098

2.866

2.711

2.599

2.514

2.447

2.393

2.348

2.278

2.203

2.124

2.074

2.039

1.994

21

4.325

3.467

3.072

2.840

2.685

2.573

2.488

2.420

2.366

2.321

2.250

2.176

2.096

2.045

2.010

1.965

22

4.301

3.443

3.049

2.817

2.661

2.549

2.464

2.397

2.342

2.297

2.226

2.151

2.071

2.020

1.984

1.938

23

4.279

3.422

3.028

2.796

2.640

2.528

2.442

2.375

2.320

2.275

2.204

2.128

2.048

1.996

1.961

1.914

24

4.260

3.403

3.009

2.776

2.621

2.508

2.423

2.355

2.300

2.255

2.183

2.108

2.027

1.975

1.939

1.892

25

4.242

3.385

2.991

2.759

2.603

2.490

2.405

2.337

2.282

2.236

2.165

2.089

2.007

1.955

1.919

1.872

26

4.225

3.369

2.975

2.743

2.587

2.474

2.388

2.321

2.265

2.220

2.148

2.072

1.990

1.938

1.901

1.853

27

4.210

3.354

2.960

2.728

2.572

2.459

2.373

2.305

2.250

2.204

2.132

2.056

1.974

1.921

1.884

1.836

28

4.196

3.340

2.947

2.714

2.558

2.445

2.359

2.291

2.236

2.190

2.118

2.041

1.959

1.906

1.869

1.820

29

4.183

3.328

2.934

2.701

2.545

2.432

2.346

2.278

2.223

2.177

2.104

2.027

1.945

1.891

1.854

1.806

30

4.171

3.316

2.922

2.690

2.534

2.421

2.334

2.266

2.211

2.165

2.092

2.015

1.932

1.878

1.841

1.792

31

4.160

3.305

2.911

2.679

2.523

2.409

2.323

2.255

2.199

2.153

2.080

2.003

1.920

1.866

1.828

1.779

32

4.149

3.295

2.901

2.668

2.512

2.399

2.313

2.244

2.189

2.142

2.070

1.992

1.908

1.854

1.817

1.767

33

4.139

3.285

2.892

2.659

2.503

2.389

2.303

2.235

2.179

2.133

2.060

1.982

1.898

1.844

1.806

1.756

34

4.130

3.276

2.883

2.650

2.494

2.380

2.294

2.225

2.170

2.123

2.050

1.972

1.888

1.833

1.795

1.745

35

4.121

3.267

2.874

2.641

2.485

2.372

2.285

2.217

2.161

2.114

2.041

1.963

1.878

1.824

1.786

1.735

36

4.113

3.259

2.866

2.634

2.477

2.364

2.277

2.209

2.153

2.106

2.033

1.954

1.870

1.815

1.776

1.726

37

4.105

3.252

2.859

2.626

2.470

2.356

2.270

2.201

2.145

2.098

2.025

1.946

1.861

1.806

1.768

1.717

38

4.098

3.245

2.852

2.619

2.463

2.349

2.262

2.194

2.138

2.091

2.017

1.939

1.853

1.798

1.760

1.708

39

4.091

3.238

2.845

2.612

2.456

2.342

2.255

2.187

2.131

2.084

2.010

1.931

1.846

1.791

1.752

1.700

40

4.085

3.232

2.839

2.606

2.449

2.336

2.249

2.180

2.124

2.077

2.003

1.924

1.839

1.783

1.744

1.693

 

Подпись: Appendix 399

Подпись: 400 Appendix

V-2/v

1

2

3

4

5

6

7

8

9

10

12

15

20

25

30

40

1 4052.181

4999.500

5403.352

5624.583

5763.650

5858.986

5928.356

5981.070

6022.473

6055.847

6106.321

6157.285

6208.730

6239.825

6260.649

6286.782

2

98.503

99.000

99.166

99.249

99.299

99.333

99.356

99.374

99.388

99.399

99.416

99.433

99.449

99.459

99.466

99.474

3

34.116

30.817

29.457

28.710

28.237

27.911

27.672

27.489

27.345

27.229

27.052

26.872

26.690

26.579

26.505

26.411

4

21.198

18.000

16.694

15.977

15.522

15.207

14.976

14.799

14.659

14.546

14.374

14.198

14.020

13.911

13.838

13.745

5

16.258

13.274

12.060

11.392

10.967

10.672

10.456

10.289

10.158

10.051

9.888

9.722

9.553

9.449

9.379

9.291

6

13.745

10.925

9.780

9.148

8.746

8.466

8.260

8.102

7.976

7.874

7.718

7.559

7.396

7.296

7.229

7.143

7

12.246

9.547

8.451

7.847

7.460

7.191

6.993

6.840

6.719

6.620

6.469

6.314

6.155

6.058

5.992

5.908

8

11.259

8.649

7.591

7.006

6.632

6.371

6.178

6.029

5.911

5.814

5.667

5.515

5.359

5.263

5.198

5.116

9

10.561

8.022

6.992

6.422

6.057

5.802

5.613

5.467

5.351

5.257

5.111

4.962

4.808

4.713

4.649

4.567

10

10.044

7.559

6.552

5.994

5.636

5.386

5.200

5.057

4.942

4.849

4.706

4.558

4.405

4.311

4.247

4.165

11

9.646

7.206

6.217

5.668

5.316

5.069

4.886

4.744

4.632

4.539

4.397

4.251

4.099

4.005

3.941

3.860

12

9.330

6.927

5.953

5.412

5.064

4.821

4.640

4.499

4.388

4.296

4.155

4.010

3.858

3.765

3.701

3.619

13

9.074

6.701

5.739

5.205

4.862

4.620

4.441

4.302

4.191

4.100

3.960

3.815

3.665

3.571

3.507

3.425

14

8.862

6.515

5.564

5.035

4.695

4.456

4.278

4.140

4.030

3.939

3.800

3.656

3.505

3.412

3.348

3.266

15

8.683

6.359

5.417

4.893

4.556

4.318

4.142

4.004

3.895

3.805

3.666

3.522

3.372

3.278

3.214

3.132

16

8.531

6.226

5.292

4.773

4.437

4.202

4.026

3.890

3.780

3.691

3.553

3.409

3.259

3.165

3.101

3.018

17

8.400

6.112

5.185

4.669

4.336

4.102

3.927

3.791

3.682

3.593

3.455

3.312

3.162

3.068

3.003

2.920

18

8.285

6.013

5.092

4.579

4.248

4.015

3.841

3.705

3.597

3.508

3.371

3.227

3.077

2.983

2.919

2.835

19

8.185

5.926

5.010

4.500

4.171

3.939

3.765

3.631

3.523

3.434

3.297

3.153

3.003

2.909

2.844

2.761

20

8.096

5.849

4.938

4.431

4.103

3.871

3.699

3.564

3.457

3.368

3.231

3.088

2.938

2.843

2.778

2.695

21

8.017

5.780

4.874

4.369

4.042

3.812

3.640

3.506

3.398

3.310

3.173

3.030

2.880

2.785

2.720

2.636

22

7.945

5.719

4.817

4.313

3.988

3.758

3.587

3.453

3.346

3.258

3.121

2.978

2.827

2.733

2.667

2.583

23

7.881

5.664

4.765

4.264

3.939

3.710

3.539

3.406

3.299

3.211

3.074

2.931

2.781

2.686

2.620

2.535

24

7.823

5.614

4.718

4.218

3.895

3.667

3.496

3.363

3.256

3.168

3.032

2.889

2.738

2.643

2.577

2.492

25

7.770

5.568

4.675

4.177

3.855

3.627

3.457

3.324

3.217

3.129

2.993

2.850

2.699

2.604

2.538

2.453

26

7.721

5.526

4.637

4.140

3.818

3.591

3.421

3.288

3.182

3.094

2.958

2.815

2.664

2.569

2.503

2.417

27

7.677

5.488

4.601

4.106

3.785

3.558

3.388

3.256

3.149

3.062

2.926

2.783

2.632

2.536

2.470

2.384

28

7.636

5.453

4.568

4.074

3.754

3.528

3.358

3.226

3.120

3.032

2.896

2.753

2.602

2.506

2.440

2.354

29

7.598

5.420

4.538

4.045

3.725

3.499

3.330

3.198

3.092

3.005

2.868

2.726

2.574

2.478

2.412

2.325

30

7.562

5.390

4.510

4.018

3.699

3.473

3.304

3.173

3.067

2.979

2.843

2.700

2.549

2.453

2.386

2.299

31

7.530

5.362

4.484

3.993

3.675

3.449

3.281

3.149

3.043

2.955

2.820

2.677

2.525

2.429

2.362

2.275

32

7.499

5.336

4.459

3.969

3.652

3.427

3.258

3.127

3.021

2.934

2.798

2.655

2.503

2.406

2.340

2.252

33

7.471

5.312

4.437

3.948

3.630

3.406

3.238

3.106

3.000

2.913

2.777

2.634

2.482

2.386

2.319

2.231

34

7.444

5.289

4.416

3.927

3.611

3.386

3.218

3.087

2.981

2.894

2.758

2.615

2.463

2.366

2.299

2.211

35

7.419

5.268

4.396

3.908

3.592

3.368

3.200

3.069

2.963

2.876

2.740

2.597

2.445

2.348

2.281

2.193

36

7.396

5.248

4.377

3.890

3.574

3.351

3.183

3.052

2.946

2.859

2.723

2.580

2.428

2.331

2.263

2.175

37

7.373

5.229

4.360

3.873

3.558

3.334

3.167

3.036

2.930

2.843

2.707

2.564

2.412

2.315

2.247

2.159

38

7.353

5.211

4.343

3.858

3.542

3.319

3.152

3.021

2.915

2.828

2.692

2.549

2.397

2.299

2.232

2.143

39

7.333

5.194

4.327

3.843

3.528

3.305

3.137

3.006

2.901

2.814

2.678

2.535

2.382

2.285

2.217

2.128

40

7.314

5.179

4.313

3.828

3.514

3.291

3.124

2.993

2.888

2.801

2.665

2.522

2.369

2.271

2.203

2.114

 

V

.995

.990

.975

.950

.90

.50

.10

.05

.025

.01

.005

1

0.00004

0.00016

0.00098

0.00393

0.01579

0.45494

2.70554

3.84146

5.02389

6.63490

7.87944

2

0.01003

0.02010

0.05064

0.10259

0.21072

1.38629

4.60517

5.99146

7.37776

9.21034

10.5966

3

0.07172

0.11483

0.21580

0.35185

0.58437

2.36597

6.25139

7.81473

9.34840

11.3449

12.8382

4

0.20699

0.29711

0.48442

0.71072

1.06362

3.35669

7.77944

9.48773

11.1433

13.2767

14.8603

5

0.41174

0.55430

0.83121

1.14548

1.61031

4.35146

9.23636

11.0705

12.8325

15.0863

16.7496

6

0.67573

0.87209

1.23734

1.63538

2.20413

5.34812

10.6446

12.5916

14.4494

16.8119

18.5476

7

0.98926

1.23904

1.68987

2.16735

2.83311

6.34581

12.0170

14.0671

16.0128

18.4753

20.2777

8

1.34441

1.64650

2.17973

2.73264

3.48954

7.34412

13.3616

15.5073

17.5345

20.0902

21.9550

9

1.73493

2.08790

2.70039

3.32511

4.16816

8.34283

14.6837

16.9190

19.0228

21.6660

23.5894

10

2.15586

2.55821

3.24697

3.94030

4.86518

9.34182

15.9872

18.3070

20.4832

23.2093

25.1882

11

2.60322

3.05348

3.81575

4.57481

5.57778

10.3410

17.2750

19.6751

21.9200

24.7250

26.7568

12

3.07382

3.57057

4.40379

5.22603

6.30380

11.3403

18.5493

21.0261

23.3367

26.2170

28.2995

13

3.56503

4.10692

5.00875

5.89186

7.04150

12.3398

19.8119

22.3620

24.7356

27.6882

29.8195

14

4.07467

4.66043

5.62873

6.57063

7.78953

13.3393

21.0641

23.6848

26.1189

29.1412

31.3193

15

4.60092

5.22935

6.26214

7.26094

8.54676

14.3389

22.3071

24.9958

27.4884

30.5779

32.8013

16

5.14221

5.81221

6.90766

7.96165

9.31224

15.3385

23.5418

26.2962

28.8454

31.9999

34.2672

17

5.69722

6.40776

7.56419

8.67176

10.0852

16.3382

24.7690

27.5871

30.1910

33.4087

35.7185

18

6.26480

7.01491

8.23075

9.39046

10.8649

17.3379

25.9894

28.8693

31.5264

34.8053

37.1565

19

6.84397

7.63273

8.90652

10.1170

11.6509

18.3377

27.2036

30.1435

32.8523

36.1909

38.5823

20

7.43384

8.26040

9.59078

10.8508

12.4426

19.3374

28.4120

31.4104

34.1696

37.5662

39.9968

21

8.03365

8.89720

10.2829

11.5913

13.2396

20.3372

29.6151

32.6706

35.4789

38.9322

41.4011

22

8.64272

9.54249

10.9823

12.3380

14.0415

21.3370

30.8133

33.9244

36.7807

40.2894

42.7957

23

9.26042

10.1957

11.6886

13.0905

14.8480

22.3369

32.0069

35.1725

38.0756

41.6384

44.1813

24

9.88623

10.8564

12.4012

13.8484

15.6587

23.3367

33.1962

36.4150

39.3641

42.9798

45.5585

25

10.5197

11.5240

13.1197

14.6114

16.4734

24.3366

34.3816

37.6525

40.6465

44.3141

46.9279

26

11.1602

12.1981

13.8439

15.3792

17.2919

25.3365

35.5632

38.8851

41.9232

45.6417

48.2899

27

11.8076

12.8785

14.5734

16.1514

18.1139

26.3363

36.7412

40.1133

43.1945

46.9629

49.6449

28

12.4613

13.5647

15.3079

16.9279

18.9392

27.3362

37.9159

41.3371

44.4608

48.2782

50.9934

29

13.1211

14.2565

16.0471

17.7084

19.7677

28.3361

39.0875

42.5570

45.7223

49.5879

52.3356

30

13.7867

14.9535

16.7908

18.4927

20.5992

29.3360

40.2560

43.7730

46.9792

50.8922

53.6720

31

14.4578

15.6555

17.5387

19.2806

21.4336

30.3359

41.4217

44.9853

48.2319

52.1914

55.0027

32

15.1340

16.3622

18.2908

20.0719

22.2706

31.3359

42.5847

46.1943

49.4804

53.4858

56.3281

33

15.8153

17.0735

19.0467

20.8665

23.1102

32.3358

43.7452

47.3999

50.7251

54.7755

57.6484

34

16.5013

17.7891

19.8063

21.6643

23.9523

33.3357

44.9032

48.6024

51.9660

56.0609

58.9639

35

17.1918

18.5089

20.5694

22.4650

24.7967

34.3356

46.0588

49.8018

53.2033

57.3421

60.2748

36

17.8867

19.2327

21.3359

23.2686

25.6433

35.3356

47.2122

50.9985

54.4373

58.6192

61.5812

37

18.5858

19.9602

22.1056

24.0749

26.4921

36.3355

48.3634

52.1923

55.6680

59.8925

62.8833

38

19.2889

20.6914

22.8785

24.8839

27.3430

37.3355

49.5126

53.3835

56.8955

61.1621

64.1814

39

19.9959

21.4262

23.6543

25.6954

28.1958

38.3354

50.6598

54.5722

58.1201

62.4281

65.4756

40

20.7065

22.1643

24.4330

26.5093

29.0505

39.3353

51.8051

55.7585

59.3417

63.6907

66.7660

 

Подпись: Appendix 401

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