Almon-KL estimator for the distributed lag model

Figures & data

Table 1. Estimated MSE’s when lag length is 8.

$\mathrm{\tau }$	n	${\widehat{\alpha }}_A$	${\widehat{\alpha }}^k$	${\widehat{\alpha }}^d$	${\widehat{\alpha }}^{KL}$	${\widehat{\alpha }}_A$	${\widehat{\alpha }}^k$	${\widehat{\alpha }}^d$	${\widehat{\alpha }}^{KL}$
		$\sigma =1$				$\sigma =3$
0.8	60	0.1674	0.1602	0.1650	0.1592	0.6916	0.4321	0.6715	0.4218
	100	0.0868	0.0822	0.0859	0.0817	0.3625	0.2501	0.3570	0.2471
0.9	60	0.2100	0.1839	0.2026	0.1803	1.1223	0.5994	1.0569	0.5656
	100	0.1150	0.1040	0.1126	0.1026	0.5820	0.3468	0.5644	0.3376
0.99	60	0.9213	0.5185	0.6491	0.4089	7.8423	3.2830	5.2921	2.1330
	100	0.5481	0.3218	0.4303	0.2708	4.5558	1.9046	3.4994	1.4332
0.999	60	8.1325	3.2601	2.2957	1.8764	72.8119	27.1628	19.4490	13.1906
	100	4.7814	1.9103	1.5831	1.1510	42.8105	15.4087	13.5760	7.4965
		$\sigma =5$				$\sigma =10$
0.8	60	1.7403	0.9165	1.6845	0.8882	6.6563	3.1017	6.4333	2.9896
	100	0.9136	0.5319	0.8989	0.5244	3.4961	1.7298	3.4387	1.7015
0.9	60	2.9472	1.3722	2.7656	1.2797	11.5020	4.9011	10.7757	4.5334
	100	1.5155	0.7646	1.4678	0.7408	5.8904	2.5921	5.7016	2.5005
0.99	60	21.6850	8.6932	14.5764	5.4667	86.5740	34.0261	58.0927	21.0607
	100	12.5698	4.9479	9.6402	3.6427	50.1325	19.1852	38.4314	13.9808
0.999	60	202.1728	74.9526	53.7505	35.7999	808.5568	298.9995	214.5284	141.7912
	100	118.8650	42.3641	37.5870	20.1206	475.3627	168.7284	150.1936	79.2424

Table 2. Estimated MSE’s when lag length is 16.

$\mathrm{\tau }$	n	${\widehat{\alpha }}_A$	${\widehat{\alpha }}^k$	${\widehat{\alpha }}^d$	${\widehat{\alpha }}^{KL}$	${\widehat{\alpha }}_A$	${\widehat{\alpha }}^k$	${\widehat{\alpha }}^d$	${\widehat{\alpha }}^{KL}$
		$\sigma =1$				$\sigma =3$
0.8	60	0.2962	0.2972	0.2958	0.2871	0.4960	0.4843	0.4944	0.4837
	100	0.2683	0.2671	0.2681	0.2670	0.4060	0.3863	0.4053	0.3859
0.9	60	0.3338	0.3369	0.3331	0.3268	0.6413	0.5913	0.6381	0.5900
	100	0.2507	0.2491	0.2503	0.2489	0.4353	0.4002	0.4338	0.3996
0.99	60	0.4506	0.4575	0.4395	0.4450	2.1966	1.4400	2.1064	1.3956
	100	0.3460	0.3434	0.3391	0.3415	1.4811	0.9597	1.4310	0.9361
0.999	60	2.1662	1.3504	1.6118	1.1542	17.7419	8.5903	12.5604	6.3731
	100	1.4091	0.8720	1.0823	0.7644	11.2974	5.0830	8.3108	3.8912
		$\sigma =5$				$\sigma =10$
0.8	60	0.9555	0.7870	0.9515	0.7852	3.1094	2.0455	3.0941	2.0384
	100	0.6810	0.5604	0.6793	0.5595	1.9695	1.2798	1.9630	1.2768
0.9	60	1.2570	0.9576	1.2487	0.9537	4.1441	2.5555	4.1120	2.5402
	100	0.8040	0.6159	0.8003	0.6141	2.5310	1.5152	2.5171	1.5086
0.99	60	5.6904	3.2553	5.4411	3.1313	22.0723	11.6831	21.0737	11.1863
	100	3.7503	2.0627	3.6143	1.9984	14.3853	7.1859	13.8470	6.9328
0.999	60	48.8983	23.0546	34.4455	16.7950	194.9542	90.8715	137.0058	65.6459
	100	31.0719	13.4201	22.7715	10.0711	123.7601	52.4691	90.5636	39.0175

Figure 1. Graphical illustration of the simulation result.

Table 3. Estimates and t-tests for beta coefficients.

Coeff.	Estimate	Std. Error	t value	P(>|t|)
${\beta }_0$	0.0962	0.0168	5.71	9.04E − 07
${\beta }_1$	0.123	0.0067	18.4	3.02E − 22
${\beta }_2$	0.14	0.00552	25.3	7.47E − 28
${\beta }_3$	0.146	0.00906	16.1	4.83E − 20
${\beta }_4$	0.142	0.0103	13.7	1.80E − 17
${\beta }_5$	0.127	0.00876	14.5	2.29E − 18
${\beta }_6$	0.102	0.00541	18.9	9.77E − 23
${\beta }_7$	0.0669	0.00817	8.19	2.15E − 10
${\beta }_8$	0.0214	0.0189	1.13	2.64E − 01

Table 4. Estimates for the Almon coefficients.

Coeff.	${\widehat{\alpha }}_A$	${\widehat{\alpha }}^k$	${\widehat{\alpha }}^d$	${\widehat{\alpha }}^{KL}$
${\tilde{\alpha }}_0$	73.1	1.02E − 05	1.02E − 05	1.02E − 05
${\tilde{\alpha }}_1$	0.0962	1.05E − 01	1.05E − 01	1.05E − 01
${\tilde{\alpha }}_2$	0.0320	0.02607	0.02605	0.02608
${\tilde{\alpha }}_3$	−0.0052	−4.396E − 03	−4.394E − 03	−4.397E − 03
MSE	3397.92	194.34	232.54	145.33

Table 5. Estimates and t-tests for beta coefficients.

Coeff.	Estimate	Std. Error	t value	P(>|t|)
${\beta }_0$	1.6	1.01	1.58	0.114
${\beta }_1$	0.521	0.569	0.915	0.36
${\beta }_2$	−0.491	0.67	−0.733	0.464
${\beta }_3$	−1.43	0.569	−2.52	0.0119
${\beta }_4$	−2.31	1.01	−2.28	0.0229

Table 6. Estimates for the Almon coefficients.

Coeff.	${\widehat{\alpha }}_A$	${\widehat{\alpha }}^k$	${\widehat{\alpha }}^d$	${\widehat{\alpha }}^{KL}$
Intercept	0.0134	0.0133	0.0134	0.0133
${\tilde{\alpha }}_0$	1.6010	0.6354	1.4540	0.5567
${\tilde{\alpha }}_1$	−1.1149	−0.1201	−0.9409	−0.0412
${\tilde{\alpha }}_2$	0.0345	−0.1782	−0.0044	−0.1949
MSE	2.6410	0.3690	2.0740	0.3146