Collection of Formulae and Statistical Tables for the B2-Econometrics and B3-Time Series Analysis courses and exams

Transcription

1 Collection of Formulae and Statistical Tables for the B2-Econometrics and B3-Time Series Analysis courses and exams Lars Forsberg Uppsala University Spring 2015 Abstract This collection of formulae is to be used during the courses and at the written examinations for Econometrics and Time Series Analysis. Students should, and must bring this collection of Formulae to the exam. There will be NO HANDING out of this collection at the exam location. Also note that AT THE EXAM, THERE MUST BE NO NOTES OF ANY KIND in this collection. If students have made notes in this collection during the course, they must print and bring a new copy without any notes to the exam. The sta at the exam location are instructed to randomly check the collection for notes. If they nd any notes of any kind in this collection, this will be reported to the responsible teacher, and the student will be subject to an investigation whether he or she should be reported to the disciplinary committé for cheating. Also note, that this means that students CANNOT MAKE ANY NOTES IN THIS COLLECTION DURING THE EXAM. Of course, notes made during the exam will be observationaly indistinguishable from notes made beforehand. To remedy this situation, any notes will be be assumed to be written before the exam, and thus will be reported. Moreover, students are NOT allowed to pass this collection between them at the exam. 1

2 Contents 1 Test -templete 3 2 Basic Statistics Expectation, Variance and Co-variance Regression Single linear regression Z and t-tests Con dence intervals Multiple Regression F-tests Normality test Time Series Back shift operator Di erence operator Seasonal Di erence operator Geometric series Ljung-Box Test Test of an individual correlation ADF unit root test Math Quadratic identities Power function Exponential function The logarithmic function Appendix: Statistical tables The Normal Distriubtion The Student s t-distribution The Chi-Square Distribution The F-Distribution These tables are courtesy of Dr. Thommy Perlinger, whose contribution is greatfully acknowledged. 2

3 1 Test -templete Whenever you perform a formal test on a written exam in Econometrics or Time Series Analysis, you must follow the outline in the template below for full score on that task. 1. The null and the alternative hypotheses State the null and the alternative in terms of relevant parameters. 2. Signi cance level 3. Estimator(s)/statistic(s) used State the estimator(s)/statistic(s), not nessecarily using the exact formula but it should be clear what kind of estimator(s)/statistic(s) that are being used. 4. Assumptions State the assumption(s) nessecary for the test to be valid. 5. Test statistic Write down the test statistic as a function of the estimator(s) and, if applicable, parameter value(s) under the null. State the distribution it follows (with its parameter or degrees of freedom values) under the null and your assumptions. Note: it is the statistic that follows a known distribution under the null (and the assumptions) that you should write down. This statistic might be a function of other quantities as well as estimators of the parameters in the null hypothesis. 6. Figure under the null and decision rule If it is well known, sketch the distribution of the test statistic under the null hypothesis (and your assumptions). Clearly mark out the critical value(s) and the correponding rejection region(s) and mark the probability mass in (each of) the (respective) rejection/ region(s). Clearly state the decision rule as: The null is rejected if Calculations and decision Collect all the information nessecary for the calculations, such as sample size, degrees of freedom etc. All the statistics that are being used need to be reported here, it is not su cient to refer to the estimation output. Perform the nessecary calculations or report here the nessecary numbers from the estimation output. For all calculations, rst state the "general" formula and then ll in the actual numbers. It must be perfectly clear what all numbers represent and where they come from. 8. Conclusion. Given the result and decision, write down the conclusion of the test. 3

4 2 Basic Statistics 2.1 Expectation, Variance and Co-variance Some basics, for any random variables X, Y; Z and W and constants a, b; c and d we have E (ax) = ae (X) (1) h V ar (X) = E (X E (X)) 2i (2) V ar (ax + b) = a 2 V ar (X) (3) V (ax + by ) = a 2 V (X) + b 2 V (Y ) + 2abCov (X; Y ) (4) = Cov [(ax + by ) ; (cz + dy )] (5) = accov (X; Z) + adcov (X; Y ) + bccov (Y; Z) + bdcov (Y; Y ) (6) 4

5 3 Regression 3.1 Single linear regression The single linear regression is given by Y i = X i + u i (7) The OLS-estimator of the intercept is given by c 1 = Y c 2 X and the OLS-estimator of the slope is given by c 2 = n i=1 X i X Y i Y n i=1 X i X 2 The OLS-estimator of the error term variance is c 2 = n i=1 bu2 i n k (8) The variance of the estimated slope coe cient is given by c = n i=1 X i X 2 where n is the number of observations and k is the number of coe cients in the regression equation. The coe cient of determination is given by R 2 = 1 n i=1 bu2 i n i=1 Y i Y 2 (9) 3.2 Z and t-tests To test hypotheses about the regression coe cients we can use, assuming that the error term variance is known, z Obs = b j H0 j bj N (0; 1) (10) and when we have to estimate the error term variance, we need to use t Obs = b j H0 j b bj t n k : 5

6 3.3 Con dence intervals Con dence intervals: For a normally distributed N (0; 1) variable such as z Obs = b j j N (0; 1) (11) bj consider the following probabilistic statement. Pr z =2 z Obs z =2 = 1 : (12) Con dence intervals: For a Student s t distributed variable such as t Obs = b j j t n k (13) b bj consider the following probabilistic statement. Pr t n k;=2 t Obs t n k;=2 = 1 : (14) 3.4 Multiple Regression The general multiple regression model Y i = X 2;i + ::: + 3 X k;i + u i has k coe cients, k The variance 1 regressors. 2 b j = 2 P Xj Xj R 2 j! (15) Rj 2 would be the R2 from the regression of X j upon an intercept and all the other regressors in the original model. 6

7 3.5 F-tests To test linear restrictions - note that we in this situation have two models, one unrestricted, (the original model) and one model where the restrictions of the null are imposed/implemented (the restricted model). Any linear restriction can be tested using F obs = R2 UR RR 2 =m (1 RUR 2 ) = (n k) F m;n k; (16) where RUR 2 is the coe cient of determination for the UnRestricted model R 2 R is the coe cient of determination for the Restricted model m is the number of (linear) restrictions (in the null hypothesis) n is the number of observations k is the number of regression coe cients (parameters) in the regression line of the UnRestricted model (including the intercept, if there is one). 3.6 Normality test The Jarque-Bera test for normality uses the following test statistic " # S 2 (K 3)2 JB = n (17) 6 24 (i.e. follows a chi-square distribution with two degrees of freedom) where n is the sample size S is the sample skewness K is the sample kurtosis 7

8 4 Time Series 4.1 Back shift operator Let fy t g be a stochasic process. The back-shift operator B is de ned as B 0 Y t = Y t 0 B 1 Y t = Y t 1 B 2 Y t = Y t 2 and B j Y t = Y t j : Note that for a constant c we have B j c = c: for all j: 4.2 Di erence operator Let fy t g be a stochasic process. The di erence-operator r is de ned as r 1 Y t = Y t Y t 1 also Note that for a constant c we have r j = 1 B 1 j rc = 1 B 1 1 c rc = c Bc rc = c c rc = 0: 8

9 4.3 Seasonal Di erence operator Let fy t g be a stochasic process. The seasonal di erence-operator r s is de ned as r 1 sy t = Y t Y t s We note that and Note that for a constant c we have 4.4 Geometric series r 1 s = (1 B s ) 1 r 2 s = (1 B s ) 2 r j s = (1 B s ) j rc = (1 B s ) 1 c rc = c B s c rc = c c rc = 0 For time series we might nd the following useful 1 j=0 rj = 1 1 r for jrj < 1 1 j=0 (B)j = 1 1 B for jj < 1 1 j=0 (B)j = 1 1 B for jj < 1 where B denotes the back-shift operator and and are a constants (parameters). 9

10 4.5 Ljung-Box Test The Ljung-Box test tests whether several autocorrelations (K of them) are zero simultaneously, i.e. The test statistic is given by H 0 : 1 = ::: = K = 0 H 1 : at least one j 6= 0 for j = 1; :::; K Q LB = T (T + 2) K j=1 T where T is the sample size and b j is the j:th autocorrelation, which under the null and for large samples, follows a b 2 j 2 (K p q P Q) where p; q; P and Q are the order of the AR, MA, SAR, SMA parts respectively, (assuming that we do the test on residuals from a model). That is, we subtract from K the number of parameters estimated in the model at hand. If the test is applied on raw data, of course, we have not estimated any parameters, i.e. p = q = P = Q = 0: 4.6 Test of an individual correlation To test an individual autocorrelation, we use the following test statistic H 0 : k = H0 k H 1 : k 6= H0 k z obs = b k which under the null and for large samples follows a N (0; 1) distribution and where r 1 bk = T where T is the sample size. bk H0 k j 10

11 4.7 ADF unit root test To test for a unit root in Y t we run the (auxiliary) regression and do a t-test of ry t = ay t 1 + [ 1 ry t ry t 2 + :::: + k ry t k ] + e t H 0 : a 0 H 1 : a < 0 Where the null respresents that we have a unit root in Y t ; that is, that Y t I (1) : The test statistic is the t-ratio (not really a t-ratio since it follows a nonstandard distribution under the null) ADF obs = ba 0 b ba NOTE: under the null of a unit root, the ADF obs does NOT follow any standard distribution. Its distribution is tabulated and is included in any software that has this test, e.g. Eviews. So, when performing this test, following the test-template, it is not nessecary to draw a gure of the distribution under the null and the assumptions (since that is unknown). However, it should be noted that the reason for no gure is the fact that the distribution is non-standard! The decision to reject or not - is based on the p-value of the test, compared to the chosen signi cance level of the test. 11

12 5 Math 5.1 Quadratic identities For a; b and c being real numbers, then (a + b) 2 = a 2 + b 2 + 2ab; (18) (a b) 2 = a 2 + b 2 2ab; (19) (a + b) (a b) = a 2 b 2 ; (20) and (a + b + c) 2 = a 2 + b 2 + c 2 + 2ab + 2ac + 2bc: (21) 5.2 Power function The general power function in given by f (x) = Ax r ; (22) where A and r are constants. Some rules for manipulating power functions are x r x s = x r+s (23) (xy) r = x r y r (24) r x = xr y y r (25) (x r ) s = x rs (26) x r x s = x r s (27) x 1 = 1 x (28) x n = 1 x n (29) p x = x 1=2 (30) 1 p x = x 1=2 : (31) 5.3 Exponential function For the exponential function f (x) = a x (32) 12

13 note that a is a constant and that x is the variable. We have that a x a y = a x+y (33) (a x ) y = a xy (34) 5.4 The logarithmic function De nition of ln a is given by a > 0: Now, it follows that Some rules for logarithmic function ln (ab) x = a x b x (35) a x a y = a x y (36) a b x = ax b x (37) e ln a = a (38) ln 1 = 0; (39) ln e = 1: (40) ln (xy) = ln x + ln y; (41) ln x y = ln x ln y; (42) ln x p = p ln x; (43) ln e x = x: (44) 13

14 6 Appendix: Statistical tables The Normal Distriubtion 2 These tables are courtesy of Dr. Thommy Perlinger, whose contribution is greatfully acknowledged. 14

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16 6.2 The Student s t-distribution 16

17 6.3 The Chi-Square Distribution 17

18 6.4 The F-Distribution 18

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