Springer Texts in Business and Economics

Recursive Residuals

In Section 8.1, we showed that the least squares residuals are heteroskedastic with non-zero covariances, even when the true disturbances have a scalar covariance matrix. This section studies recursive residuals which are a set of linear unbiased residuals with a scalar covariance matrix. They are independent and identically distributed when the true disturbances themselves are independent and identically distributed.2 These residuals are natural in time-series regressions and can be constructed as follows:

1. Choose the first t > k observations and compute f3t = (X'tXt)-1X'tYt where Xt denotes the t x k matrix of t observations on k variables and Yt' = (y1,...,yt). The recursive residuals are basically standardized one-step ahead forecast residuals:

Table 8.2 Diagnostic Statistics for the Cigarettes Example

OBS	STATE	LNC	LNP	LNY	PREDICTED	e	e	e*	Cook's D	Leverage	DFFITS	COVRATIO
1	AL	4.96213	0.20487	4.64039	4.8254	0.1367	0.857	0.8546	0.012	0.0480	0.1919	1.0704
2	AZ	4.66312	0.16640	4.68389	4.8844	-0.2213	-1.376	-1.3906	0.021	0.0315	-0.2508	0.9681
3	AR	5.10709	0.23406	4.59435	4.7784	0.3287	2.102	2.1932	0.136	0.0847	0.6670	0.8469
4	CA	4.50449	0.36399	4.88147	4.6540	-0.1495	-0.963	-0.9623	0.033	0.0975	-0.3164	1.1138
5	CT	4.66983	0.32149	5.09472	4.7477	-0.0778	-0.512	-0.5077	0.014	0.1354	-0.2009	1.2186
6	DE	5.04705	0.21929	4.87087	4.8458	0.2012	1.252	1.2602	0.018	0.0326	0.2313	0.9924
7	DC	4.65637	0.28946	5.05960	4.7845	-0.1281	-0.831	-0.8280	0.029	0.1104	-0.2917	1.1491
8	FL	4.80081	0.28733	4.81155	4.7446	0.0562	0.352	0.3482	0.002	0.0431	0.0739	1.1118
9	GA	4.97974	0.12826	4.73299	4.9439	0.0358	0.224	0.2213	0.001	0.0402	0.0453	1.1142
10	ID	4.74902	0.17541	4.64307	4.8653	-0.1163	-0.727	-0.7226	0.008	0.0413	-0.1500	1.0787
11	IL	4.81445	0.24806	4.90387	4.8130	0.0014	0.009	0.0087	0.000	0.0399	0.0018	1.1178
12	IN	5.11129	0.08992	4.72916	4.9946	0.1167	0.739	0.7347	0.013	0.0650	0.1936	1.1046
13	IA	4.80857	0.24081	4.74211	4.7949	0.0137	0.085	0.0843	0.000	0.0310	0.0151	1.1070
14	KS	4.79263	0.21642	4.79613	4.8368	-0.0442	-0.273	-0.2704	0.001	0.0223	-0.0408	1.0919
15	KY	5.37906	-0.03260	4.64937	5.1448	0.2343	1.600	1.6311	0.210	0.1977	0.8098	1.1126
16	LA	4.98602	0.23856	4.61461	4.7759	0.2101	1.338	1.3504	0.049	0.0761	0.3875	1.0224
17	ME	4.98722	0.29106	4.75501	4.7298	0.2574	1.620	1.6527	0.051	0.0553	0.4000	0.9403
18	MD	4.77751	0.12575	4.94692	4.9841	-0.2066	-1.349	-1.3624	0.084	0.1216	-0.5070	1.0731
19	MA	4.73877	0.22613	4.99998	4.8590	-0.1202	-0.769	-0.7653	0.018	0.0856	-0.2341	1.1258
20	MI	4.94744	0.23067	4.80620	4.8195	0.1280	0.792	0.7890	0.005	0.0238	0.1232	1.0518
21	MN	4.69589	0.34297	4.81207	4.6702	0.0257	0.165	0.1627	0.001	0.0864	0.0500	1.1724
22	MS	4.93990	0.13638	4.52938	4.8979	0.0420	0.269	0.2660	0.002	0.0883	0.0828	1.1712
23	MO	5.06430	0.08731	4.78189	5.0071	0.0572	0.364	0.3607	0.004	0.0787	0.1054	1.1541
24	MT	4.73313	0.15303	4.70417	4.9058	-0.1727	-1.073	-1.0753	0.012	0.0312	-0.1928	1.0210
25	NE	4.77558	0.18907	4.79671	4.8735	-0.0979	-0.607	-0.6021	0.003	0.0243	-0.0950	1.0719
26	NV	4.96642	0.32304	4.83816	4.7014	0.2651	1.677	1.7143	0.065	0.0646	0.4504	0.9366
27	NH	5.10990	0.15852	5.00319	4.9500	0.1599	1.050	1.0508	0.055	0.1308	0.4076	1.1422
28	NJ	4.70633	0.30901	5.10268	4.7657	-0.0594	-0.392	-0.3879	0.008	0.1394	-0.1562	1.2337
29	NM	4.58107	0.16458	4.58202	4.8693	-0.2882	-1.823	-1.8752	0.076	0.0639	-0.4901	0.9007
30	NY	4.66496	0.34701	4.96075	4.6904	-0.0254	-0.163	-0.1613	0.001	0.0888	-0.0503	1.1755
31	ND	4.58237	0.18197	4.69163	4.8649	-0.2825	-1.755	-1.7999	0.031	0.0295	-0.3136	0.8848
32	OH	4.97952	0.12889	4.75875	4.9475	0.0320	0.200	0.1979	0.001	0.0423	0.0416	1.1174
33	OK	4.72720	0.19554	4.62730	4.8356	-0.1084	-0.681	-0.6766	0.008	0.0505	-0.1560	1.0940
34	PA	4.80363	0.22784	4.83516	4.8282	-0.0246	-0.153	-0.1509	0.000	0.0257	-0.0245	1.0997
35	RI	4.84693	0.30324	4.84670	4.7293	0.1176	0.738	0.7344	0.010	0.0504	0.1692	1.0876
36	SC	5.07801	0.07944	4.62549	4.9907	0.0873	0.555	0.5501	0.008	0.0725	0.1538	1.1324
37	SD	4.81545	0.13139	4.67747	4.9301	-0.1147	-0.716	-0.7122	0.007	0.0402	-0.1458	1.0786
38	TN	5.04939	0.15547	4.72525	4.9062	0.1432	0.890	0.8874	0.008	0.0294	0.1543	1.0457
39	TX	4.65398	0.28196	4.73437	4.7384	-0.0845	-0.532	-0.5271	0.005	0.0546	-0.1267	1.1129
40	UT	4.40859	0.19260	4.55586	4.8273	-0.4187	-2.679	-2.9008	0.224	0.0856	-0.8876	0.6786
41	VT	5.08799	0.18018	4.77578	4.8818	0.2062	1.277	1.2869	0.014	0.0243	0.2031	0.9794
42	VA	4.93065	0.11818	4.85490	4.9784	-0.0478	-0.304	-0.3010	0.003	0.0773	-0.0871	1.1556
43	WA	4.66134	0.35053	4.85645	4.6677	-0.0064	-0.041	-0.0404	0.000	0.0866	-0.0124	1.1747
44	WV	4.82454	0.12008	4.56859	4.9265	-0.1020	-0.647	-0.6429	0.011	0.0709	-0.1777	1.1216
45	WI	4.83026	0.22954	4.75826	4.8127	0.0175	0.109	0.1075	0.000	0.0254	0.0174	1.1002
46	WY	5.00087	0.10029	4.71169	4.9777	0.0232	0.146	0.1444	0.000	0.0555	0.0350	1.1345

Table 8.3 Regression of Real Per-Capita Consumption of Cigarettes

Dep Obs	Var LNC	Predict Value	Std Err Predict	Lower95% Mean	Upper95% Mean	Lower95% Predict	Upper95% Predict	Std Err Residual	Student Residual	Residual -	to 1 о	2 Cook’s D
і	4.9621	4.8254	0.0.36	4.75.32	4.8976	4.4880	5.1628	0.1.367	0.159	0.857		к	0.012
2	4.6631	4.8844	0.029	4.8259	4.9429	4.5497	5.2191	-0.221.3	0.161	-1.376	к к		0.021
3	5.1071	4.7784	0.048	4.6825	4.874.3	4.4.351	5.1217	0.3287	0.156	2.102		ккк к	0.1.36
4	4.5045	4.6540	0.051	4.5511	4.7570	4.3087	4.999.3	-0.1495	0.155	-0.96.3	к		0.0.33
5	4.6698	4.7477	0.060	4.6264	4.8689	4.3965	5.0989	-0.0778	0.152	-0.512	к		0.014
6	5.0471	4.8458	0.0.30	4.786.3	4.905.3	4.5109	5.1808	0.2012	0.161	1.252		к к	0.018
7	4.6564	4.7845	0.054	4.6750	4.8940	4.4.372	5.1.318	-0.1281	0.154	-0.8.31	к		0.029
8	4.8008	4.7446	0.0.34	4.6761	4.81.30	4.4079	5.0812	0.0562	0.160	0.352			0.002
9	4.9797	4.94.39	0.0.33	4.8778	5.0100	4.6078	5.2801	0.0.358	0.160	0.224			0.001
10	4.7490	4.865.3	0.0.33	4.798.3	4.9.32.3	4.5290	5.2016	-0.116.3	0.160	-0.727	к		0.008
11	4.8145	4.81.30	0.0.33	4.7472	4.8789	4.4769	5.1491	0.00142	0.160	0.009			0.000
12	5.111.3	4.9946	0.042	4.9106	5.0786	4.6544	5.3.347	0.1167	0.158	0.7.39		к	0.01.3
13	4.8086	4.7949	0.029	4.7.368	4.8529	4.4602	5.1295	0.01.37	0.161	0.085			0.000
14	4.7926	4.8.368	0.024	4.7876	4.8860	4.50.36	5.1701	-0.0442	0.162	-0.27.3			0.001
15	5.3791	5.1448	0.07.3	4.9982	5.291.3	4.7841	5.5055	0.2.34.3	0.146	1.600		ккк	0.210
16	4.9860	4.7759	0.045	4.6850	4.8668	4.4.340	5.1178	0.2101	0.157	1.3.38		к к	0.049
17	4.9872	4.7298	0.0.38	4.652.3	4.8074	4.3912	5.0684	0.2574	0.159	1.620		ккк	0.051
18	4.7775	4.9841	0.057	4.8692	5.0991	4.6.351	5.3.332	-0.2066	0.15.3	-1.349	к к		0.084
19	4.7.388	4.8590	0.048	4.7625	4.9554	4.5155	5.2024	-0.1202	0.156	-0.769	к		0.018
20	4.9474	4.8195	0.025	4.7686	4.870.3	4.4860	5.15.30	0.1280	0.161	0.792		к	0.005
21	4.6959	4.6702	0.048	4.57.3.3	4.7671	4.3267	5.01.37	0.0257	0.156	0.165			0.001
22	4.9.399	4.8979	0.049	4.8000	4.9959	4.5541	5.2418	0.0420	0.156	0.269			0.002
23	5.064.3	5.0071	0.046	4.9147	5.0996	4.6648	5.3495	0.0572	0.157	0.364			0.004
24	4.7.3.31	4.9058	0.029	4.8476	4.9640	4.5711	5.2405	-0.1727	0.161	-1.07.3	к к		0.012
25	4.7756	4.87.35	0.025	4.8221	4.9249	4.5.399	5.2071	-0.0979	0.161	-0.607	к		0.00.3
26	4.9664	4.7014	0.042	4.6176	4.7851	4.361.3	5.0414	0.2651	0.158	1.677		ккк	0.065
27	5.1099	4.9500	0.059	4.8.308	5.0692	4.5995	5.3005	0.1599	0.152	1.050		к к	0.055
28	4.706.3	4.7657	0.061	4.6427	4.8888	4.41.39	5.1176	-0.0594	0.152	-0.392			0.008
29	4.5811	4.869.3	0.041	4.7859	4.9526	4.529.3	5.2092	-0.2882	0.158	-1.82.3	к к к		0.076
30	4.6650	4.6904	0.049	4.5922	4.7886	4.3465	5.0.34.3	-0.0254	0.156	-0.16.3			0.001
31	4.5824	4.8649	0.028	4.808.3	4.9215	4.5.305	5.199.3	-0.2825	0.161	-1.755	к к к		0.0.31
32	4.9795	4.9475	0.0.34	4.8797	5.015.3	4.6110	5.2840	0.0.320	0.160	0.200			0.001
33	4.7272	4.8.356	0.0.37	4.7616	4.9097	4.4978	5.17.35	-0.1084	0.159	-0.681	к		0.008
34	4.80.36	4.8282	0.026	4.7754	4.8811	4.4944	5.1621	-0.0246	0.161	-0.15.3			0.000
35	4.8469	4.729.3	0.0.37	4.655.3	4.80.3.3	4.3915	5.0671	0.1176	0.159	0.7.38		к	0.010
36	5.0780	4.9907	0.044	4.9020	5.0795	4.6494	5.3.320	0.087.3	0.157	0.555		к	0.008
37	4.8155	4.9.301	0.0.33	4.8640	4.996.3	4.5940	5.266.3	-0.1147	0.160	-0.716	к		0.007
38	5.0494	4.9062	0.028	4.8497	4.9626	4.5718	5.2406	0.14.32	0.161	0.890		к	0.008
39	4.6540	4.7.384	0.0.38	4.6614	4.8155	4.4000	5.0769	-0.0845	0.159	-0.5.32	к		0.005
40	4.4086	4.827.3	0.048	4.7.308	4.92.37	4.48.39	5.1707	-0.4187	0.156	-2.679	к к ккк		0.224
41	5.0880	4.8818	0.025	4.8.304	4.9.3.32	4.5482	5.2154	0.2062	0.161	1.277		к к	0.014
42	4.9.307	4.9784	0.045	4.8868	5.0701	4.6.36.3	5.3205	-0.0478	0.157	-0.304			0.00.3
43	4.661.3	4.6677	0.048	4.5708	4.7647	4.3242	5.011.3	-0.006.38	0.156	-0.041			0.000
44	4.8245	4.9265	0.044	4.8.387	5.014.3	4.5854	5.2676	-0.1020	0.158	-0.647	к		0.011
45	4.8.30.3	4.8127	0.026	4.7602	4.865.3	4.4790	5.1465	0.0175	0.161	0.109			0.000
46	5.0009	4.9777	0.0.39	4.9000	5.055.3	4.6.391	5.316.3	0.02.32	0.159	0.146			0.000

Sum of Residuals 0

Sum of Squared Residuals 1.1485

Predicted Resid SS (Press) 1.3406

2. Add the (t + 1)-th observation to the data and obtain i3t+l Compute wt+2.

Repeat step 2, adding one observation at a time. In time-series regressions, one usually starts with the first ^-observations and obtain (T — k) forward recursive residuals. These recursive residuals can be computed using the updating formula given in (8.11) with A = (X'tXt) and a = —b = x't+i. Therefore,

and only (XtX^ i have to be computed. Also,

Подпись: (8.32) &+1 = & + (XtXt) ixt+i(yt+i — x't+iPt)//t+i where /t+i = 1 + xt+i(Xt/Xt)-ixt+i, see problem 13.

Alternatively, one can compute these residuals by regressing Yt+i on Xt+i and dt+i where dt+i = 1 for the (t + 1)-th observation, and zero otherwise, see equation (8.5). The estimated coefficient of dt+i is the numerator of wt+i. The standard error of this estimate is s^ times the denominator of wt+i, where s^ is the standard error of this regression. Hence, wt+i can be retrieved as s^ multiplied by the t-statistic corresponding to dt+i. This computation has to be performed sequentially, in each case generating the corresponding recursive residual. This may be computationally inefficient, but it is simple to generate using regression packages.

It is obvious from (8.30) that if ut ~ IIN(0, a2), then wt+i has zero mean and var(wt+i) = a2. Furthermore, wt+i is linear in the y’s. Therefore, it is normally distributed. It remains to show that the recursive residuals are independent. Given normality, it is sufficient to show that

cov(wt+i, ws+i) = 0 for t = s; t, s = k,...,T — 1

This is left as an exercise for the reader, see problem 13.

Alternatively, one can express the T — k vector of recursive residuals as w = Cy where C is of dimension (T — k) x T as follows:

xtT (XT— iXT — i) iX'T-i i

V fT V fT

Problem 14 asks the reader to verify that w = Cy, using (8.30). Also, that the matrix C satisfies the following properties:

Подпись: (8.35) (i) CX = 0 (ii) CCC = It-k (iii) C'C = Px

This means that the recursive residuals w are (LUS) linear in y, unbiased with mean zero and have a scalar variance-covariance matrix: var(w) = CE(uu')C' = а2Іт-к. Property (iii) also
means that w'w = y'C'Cy = y'PxУ = e'e. This means that the sum of squares of (T — k) recursive residuals is equal to the sum of squares of T least squares residuals. One can also show from (8.32) that

RSSt+i = RSSt + w2t+l for t = k,...,T — 1 (8.36)

where RSSt = (Yt — Xtf3t)'(Yt — Xt/3t), see problem 14. Note that for t = k; RSS = 0, since with k observations one gets a perfect fit and zero residuals. Therefore

RSSt = £Tt=k+l wt = £T=1 e2 (8.37)

Springer Texts in Business and Economics

The General Linear Model: The Basics

7.1 Invariance of the fitted values and residuals to non-singular transformations of the independent variables. The regression model in (7.1) can be written as y = XCC-1" + u where …

Regression Diagnostics and Specification Tests

8.1 Since H = PX is idempotent, it is positive semi-definite with b0H b > 0 for any arbitrary vector b. Specifically, for b0 = (1,0,.., 0/ we get hn …

Generalized Least Squares

9.1 GLS Is More Efficient than OLS. a. Equation (7.5) of Chap. 7 gives "ois = " + (X'X)-1X'u so that E("ois) = " as long as X and u …

Recursive Residuals

Springer Texts in Business and Economics

The General Linear Model: The Basics

Regression Diagnostics and Specification Tests

Generalized Least Squares

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