Appendix E: Equations | RDP 2019-01: Online Appendices: A Model of the Australian Housing Market

RDP 2019-01: Online Appendices: A Model of the Australian Housing Market Appendix E: Equations

Trent Saunders and Peter Tulip

March 2019

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E.1 Estimated Equations

E1: Detached housing approvals (constant prices)

Signal equation: dlog(bahousevol)-@mean(dlog(bahousevol)) = c(1)*(log(bahousevol(−1)/rinc(−1))) + c(3)*(dlog(bahousevol(−1))-@mean(dlog(bahousevol(−1)))) + c(4)*(dlog(rdp(−1))-@mean(dlog(rdp(−1)))) + c(6)*gst + c(7)*gst(−1) + c(8)*gst(−2) + c(9)*(d(rvmr(−1))) + c(10)*(d(rvmr(−2))) + c(11)*(d(rvmr(−3))) + sv1 + e1

State Equation: sv1 = sv1(−1) + e2

Sspace: SS_HOUSE
Method: Maximum likelihood (BFGS / Marquardt steps)
Date: 12/21/18 Time: 14:14
Sample: 1987Q1 2018Q2
Included observations: 126
User prior mean: MPRIOR_HOUSE
User prior variance: VPRIOR_HOUSE
Convergence achieved after 26 iterations
Coefficient covariance computed using outer product of gradients

	Coefficient	Std. Error	z-Statistic	Prob.
C(1)	−0.100361	0.048104	−2.086352	0.0369
C(3)	0.193319	0.091692	2.108346	0.0350
C(4)	0.696450	0.327964	2.123553	0.0337
C(6)	−0.292142	0.302617	−0.965387	0.3344
C(7)	−0.332333	0.290471	−1.144118	0.2526
C(8)	0.125014	0.276530	0.452083	0.6512
C(9)	−0.040479	0.008707	−4.648956	0.0000
C(10)	−0.016503	0.012830	−1.286257	0.1984
C(11)	−0.004935	0.012121	−0.407153	0.6839
C(111)	−0.040104	0.002944	−13.62391	0.0000
C(112)	−0.003758	0.002652	−1.416771	0.1565
	Final State	Root MSE	z-Statistic	Prob.
SV1	0.106484	0.012567	8.473480	0.0000
Log likelihood	219.1583	Akaike info criterion		−3.304100
Parameters	11	Schwarz criterion		−3.056488
Diffuse priors	0	Hannan-Quinn criter.		−3.203503

E2: Higher-density housing approvals (chain volume)

Signal equation: dlog(baothervol)-@mean(dlog(baothervol)) = c(1)*(log(baothervol(−1)/rinc(−1))) + c(3)*(dlog(baothervol(−1))-@mean(dlog(baothervol(−1)))) + c(4)*(dlog(baothervol(−2))-@mean(dlog(baothervol(−2)))) + c(5)*gst(−1) + c(6)*gst + c(7)*(dlog(rdp(−1))-@mean(dlog(rdp(−1)))) + c(8)*(dlog(rdp(−2))-@mean(dlog(rdp(−2)))) + c(11)*d(rvmr(−4)) + c(12)*d(rvmr(−5)) + c(13)*d(rvmr(−6)) + sv1 + e1

State Equation: sv1 = sv1(−1) + e2

Sspace: SS_OTHER
Method: Maximum likelihood (BFGS / Marquardt steps)
Date: 12/21/18 Time: 14:15
Sample: 1987Q1 2018Q2
Included observations: 126
User prior mean: MPRIOR_OTHER
User prior variance: VPRIOR_OTHER
Convergence achieved after 21 iterations
Coefficient covariance computed using outer product of gradients

	Coefficient	Std. Error	z-Statistic	Prob.
C(1)	−0.218119	0.142448	−1.531222	0.1257
C(3)	−0.359422	0.118268	−3.039046	0.0024
C(4)	−0.235235	0.099542	−2.363170	0.0181
C(5)	−0.352351	2.084274	−0.169052	0.8658
C(6)	−0.134336	2.282680	−0.058850	0.9531
C(7)	2.865936	1.198789	2.390693	0.0168
C(8)	0.135682	1.277500	0.106209	0.9154
C(11)	−0.047596	0.040156	−1.185266	0.2359
C(12)	−0.019440	0.039047	−0.497867	0.6186
C(13)	−0.040988	0.036529	−1.122076	0.2618
C(111)	−0.127044	0.010181	−12.47890	0.0000
C(112)	0.012171	0.013181	0.923346	0.3558
	Final State	Root MSE	z-Statistic	Prob.
SV1	0.176248	0.040275	4.376168	0.0000
Log likelihood	74.64281	Akaike info criterion		−0.994330
Parameters	12	Schwarz criterion		−0.724208
Diffuse priors	0	Hannan-Quinn criter.		−0.884588

E3: Alterations and additions approvals (chain volume)

Signal equation: dlog((baaavol))-@mean(dlog((baaavol))) = c(1)*(log((baaavol(−1))/rinc(−1))) + c(2)*(dlog((baaavol(−1)))-@mean(dlog((baaavol(−1))))) + c(4)*(dlog(rdp(−1))-@mean(dlog(rdp(−1)))) + c(5)*(dlog(rdp(−2))-@mean(dlog(rdp(−2)))) + c(6)*gst + c(7)*gst(−1) + c(8)*d(rvmr(−1))+ c(9)*d(rvmr(−2)) + c(10)*d(rvmr(−3)) + sv1 + e1

State Equation: sv1 = sv1(−1) + e2

Sspace: SS_AA
Method: Maximum likelihood (BFGS / Marquardt steps)
Date: 12/21/18 Time: 14:15
Sample: 1987Q1 2018Q2
Included observations: 126
User prior mean: MPRIOR_AA
User prior variance: VPRIOR_AA
Convergence achieved after 23 iterations
Coefficient covariance computed using outer product of gradients

	Coefficient	Std. Error	z-Statistic	Prob.
C(1)	−0.182937	0.096898	−1.887927	0.0590
C(2)	−0.235186	0.092095	−2.553729	0.0107
C(4)	0.540372	0.355093	1.521776	0.1281
C(5)	0.237203	0.409203	0.579671	0.5621
C(6)	−0.094603	0.406522	−0.232712	0.8160
C(7)	−0.297929	0.357951	−0.832317	0.4052
C(8)	−0.010476	0.012883	−0.813157	0.4161
C(9)	−0.012689	0.015948	−0.795629	0.4262
C(10)	−0.013542	0.015700	−0.862554	0.3884
C(111)	−0.045972	0.003840	−11.97026	0.0000
C(112)	0.008298	0.004553	1.822503	0.0684
	Final State	Root MSE	z-Statistic	Prob.
SV1	0.175294	0.020432	8.579404	0.0000
Log likelihood	196.5964	Akaike info criterion		−2.945975
Parameters	11	Schwarz criterion		−2.698363
Diffuse priors	0	Hannan-Quinn criter.		−2.845378

E4: Detached housing commencements (number)

Dependent Variable: DLOG(CHOUSENO) Method: Least Squares
Date: 12/21/18 Time: 14:15
Sample: 1988Q1 2018Q2
Included observations: 122

Variable	Coefficient	Std. Error	t-Statistic	Prob.
C	−0.029263	0.003474	−8.423945	0.0000
LOG(BAHOUSENO(−1))-LOG(CHOUSENO(−1))	0.942813	0.067582	13.95060	0.0000
DLOG(BAHOUSENO)	0.500654	0.048050	10.41949	0.0000
GST	0.037435	0.031027	1.206553	0.2300
GST(−1)	−0.151889	0.032909	−4.615442	0.0000
R-squared	0.875946	Mean dependent var		0.001504
Adjusted R-squared	0.871705	S.D. dependent var		0.079474
S.E. of regression	0.028466	Akaike info criterion		−4.240084
Sum squared resid	0.094807	Schwarz criterion		−4.125165
Log likelihood	263.6451	Hannan-Quinn criter.		−4.193407
F-statistic	206.5345	Durbin-Watson stat		1.968129
Prob(F-statistic)	0.000000

E5: Higher-density housing commencements (number)

Dependent Variable: DLOG(COTHERNO)
Method: Least Squares
Date: 12/21/18 Time: 14:15
Sample: 1988Q1 2018Q2
Included observations: 122

Variable	Coefficient	Std. Error	t-Statistic	Prob.
C	−0.068396	0.009727	−7.031357	0.0000
LOG(BAOTHERNO(−1))-LOG(COTHERNO(−1))	0.872102	0.078301	11.13781	0.0000
DLOG(BAOTHERNO)	0.555883	0.055985	9.929168	0.0000
GST(−1)	−0.115966	0.076289	−1.520094	0.1312
R-squared	0.617786	Mean dependent var		0.011185
Adjusted R-squared	0.608069	S.D. dependent var		0.118717
S.E. of regression	0.074322	Akaike info criterion		−2.328587
Sum squared resid	0.651800	Schwarz criterion		−2.236652
Log likelihood	146.0438	Hannan-Quinn criter.		−2.291246
F-statistic	63.57588	Durbin-Watson stat		1.987865
Prob(F-statistic)	0.000000

E6: Detached housing completions (number)

Dependent Variable: DLOG(COMPHOUSENO)
Method: Least Squares
Date: 12/21/18 Time: 14:15
Sample: 1988Q1 2018Q2
Included observations: 122

Variable	Coefficient	Std. Error	t-Statistic	Prob.
C	−0.003440	0.003754	−0.916198	0.3614
LOG(COMPHOUSENO(−1))-LOG(CHOUSENO(−1))	−0.413561	0.037358	−11.07036	0.0000
GST	0.084368	0.042668	1.977293	0.0504
GST(−1)	−0.092786	0.049826	−1.862187	0.0651
DLOG(CHOUSENO)	0.125903	0.057978	2.171558	0.0319
R-squared	0.592003	Mean dependent var		0.002075
Adjusted R-squared	0.578055	S.D. dependent var		0.062532
S.E. of regression	0.040619	Akaike info criterion		−3.529051
Sum squared resid	0.193037	Schwarz criterion		−3.414132
Log likelihood	220.2721	Hannan-Quinn criter.		−3.482375
F-statistic	42.44173	Durbin-Watson stat		2.580016
Prob(F-statistic)	0.000000

E7: Higher-density housing completions (number)

Dependent Variable: DLOG(COMPOTHERNO)
Method: Least Squares
Date: 12/21/18 Time: 14:15
Sample: 1988Q1 2018Q2
Included observations: 122

Variable	Coefficient	Std. Error	t-Statistic	Prob.
C	−0.017681	0.009958	−1.775475	0.0785
LOG(COMPOTHERNO(−1))-LOG(COTHERNO(−1))	−0.360375	0.053542	−6.730742	0.0000
GST	0.254086	0.098086	2.590426	0.0108
GST(−1)	−0.002845	0.101371	−0.028067	0.9777
DLOG(COTHERNO)	0.130662	0.076709	1.703351	0.0912
DLOG(COTHERNO(−1))	−0.246437	0.085383	−2.886243	0.0047
DLOG(COMPOTHERNO(−1))	−0.175762	0.078136	−2.249434	0.0264
R-squared	0.365085	Mean dependent var		0.011867
Adjusted R-squared	0.331959	S.D. dependent var		0.118004
S.E. of regression	0.096449	Akaike info criterion		−1.783936
Sum squared resid	1.069781	Schwarz criterion		−1.623050
Log likelihood	115.8201	Hannan-Quinn criter.		−1.718589
F-statistic	11.02108	Durbin-Watson stat		2.133582
Prob(F-statistic)	0.000000

E8: Detached housing commencements (chain volume)

Dependent Variable: DLOG(CHOUSEVOL)
Method: Least Squares
Date: 12/21/18 Time: 14:15
Sample: 1988Q1 2018Q2
Included observations: 122

Variable	Coefficient	Std. Error	t-Statistic	Prob.
C	−0.002494	0.002707	−0.921417	0.3587
LOG(BAHOUSEVOL(−1))-LOG(CHOUSEVOL(−1))	0.849068	0.065621	12.93905	0.0000
DLOG(BAHOUSEVOL)	0.480369	0.044405	10.81785	0.0000
GST(−1)	−0.177607	0.034742	−5.112143	0.0000
R-squared	0.870611	Mean dependent var		0.003876
Adjusted R-squared	0.867322	S.D. dependent var		0.079932
S.E. of regression	0.029115	Akaike info criterion		−4.202883
Sum squared resid	0.100027	Schwarz criterion		−4.110948
Log likelihood	260.3759	Hannan-Quinn criter.		−4.165542
F-statistic	264.6599	Durbin-Watson stat		1.887565
Prob(F-statistic)	0.000000

E9: Higher-density housing commencements (chain volume)

Dependent Variable: DLOG(COTHERVOL)
Method: Least Squares
Date: 12/21/18 Time: 14:15
Sample: 1988Q1 2018Q2
Included observations: 122

Variable	Coefficient	Std. Error	t-Statistic	Prob.
C	0.023119	0.010038	2.303231	0.0231
LOG(COTHERVOL(−1))-LOG(BAOTHERVOL(−1))	−0.850425	0.152079	−5.592015	0.0000
DLOG(BAOTHERVOL)	0.521388	0.065561	7.952758	0.0000
DLOG(BAOTHERVOL(−1))	0.045953	0.123995	0.370607	0.7116
DLOG(BAOTHERVOL(−2))	−0.001397	0.090965	−0.015363	0.9878
DLOG(COTHERVOL(−1))	−0.115555	0.126230	−0.915439	0.3619
DLOG(COTHERVOL(−2))	−0.079271	0.085163	−0.930817	0.3539
GST(−1)	0.013992	0.108521	0.128935	0.8976
R-squared	0.584403	Mean dependent var		0.018657
Adjusted R-squared	0.558884	S.D. dependent var		0.157234
S.E. of regression	0.104429	Akaike info criterion		−1.617292
Sum squared resid	1.243219	Schwarz criterion		−1.433422
Log likelihood	106.6548	Hannan-Quinn criter.		−1.542610
F-statistic	22.90064	Durbin-Watson stat		1.990594
Prob(F-statistic)	0.000000

E10: Alterations and additions commencements (chain volume)

Dependent Variable: DLOG(CAAVOL)
Method: Least Squares
Date: 12/21/18 Time: 14:15
Sample: 1988Q1 2018Q2
Included observations: 122

Variable	Coefficient	Std. Error	t-Statistic	Prob.
C	0.034945	0.005187	6.736545	0.0000
LOG(CAAVOL(−1))-LOG(BAAAVOL(−1))	−0.691079	0.078361	−8.819172	0.0000
DLOG(BAAAVOL)	0.555305	0.068310	8.129244	0.0000
DLOG(BAAAVOL(−1))	−0.020150	0.067270	−0.299545	0.7651
GST(−1)	−0.261513	0.047196	−5.540980	0.0000
R-squared	0.699772	Mean dependent var		0.002968
Adjusted R-squared	0.689508	S.D. dependent var		0.074833
S.E. of regression	0.041699	Akaike info criterion		−3.476581
Sum squared resid	0.203436	Schwarz criterion		−3.361662
Log likelihood	217.0714	Hannan-Quinn criter.		−3.429904
F-statistic	68.17600	Durbin-Watson stat		2.147701
Prob(F-statistic)	0.000000

E11: Detached housing commencements (public, chain volume)

Dependent Variable: DLOG(CHOUSEVOL_PUBLIC)
Method: Least Squares
Date: 12/21/18 Time: 14:15
Sample: 1988Q1 2018Q2
Included observations: 122

Variable	Coefficient	Std. Error	t-Statistic	Prob.
C	1.445465	0.411947	3.508864	0.0006
LOG(CHOUSEVOL_PUBLIC(−1))	−0.300257	0.084934	−3.535183	0.0006
DLOG(CHOUSEVOL_PUBLIC(−1))	−0.273546	0.104267	−2.623514	0.0099
DLOG(CHOUSEVOL_PUBLIC(−2))	−0.003812	0.104279	−0.036556	0.9709
DLOG(CHOUSEVOL_PUBLIC(−3))	0.050814	0.091240	0.556925	0.5786
R-squared	0.264975	Mean dependent var		−0.005281
Adjusted R-squared	0.239846	S.D. dependent var		0.277291
S.E. of regression	0.241762	Akaike info criterion		0.038390
Sum squared resid	6.838492	Schwarz criterion		0.153309
Log likelihood	2.658185	Hannan-Quinn criter.		0.085067
F-statistic	10.54455	Durbin-Watson stat		1.995702
Prob(F-statistic)	0.000000

E12: Higher-density housing commencements (public, chain volume)

Dependent Variable: DLOG(COTHERVOL_PUBLIC)
Method: Least Squares
Date: 12/21/18 Time: 14:15
Sample (adjusted): 1988Q3 2018Q2
Included observations: 120 after adjustments

Variable	Coefficient	Std. Error	t-Statistic	Prob.
C	1.137847	0.523628	2.173009	0.0318
LOG(COTHERVOL_PUBLIC(−1))	−0.243867	0.108272	−2.252360	0.0262
DLOG(COTHERVOL_PUBLIC(−1))	−0.486876	0.116198	−4.190062	0.0001
DLOG(COTHERVOL_PUBLIC(−2))	−0.313617	0.113914	−2.753106	0.0069
DLOG(COTHERVOL_PUBLIC(−3))	−0.381888	0.092696	−4.119792	0.0001
R-squared	0.418509	Mean dependent var		−0.026037
Adjusted R-squared	0.398283	S.D. dependent var		0.881506
S.E. of regression	0.683788	Akaike info criterion		2.118435
Sum squared resid	53.77002	Schwarz criterion		2.234580
Log likelihood	−122.1061	Hannan-Quinn criter.		2.165602
F-statistic	20.69185	Durbin-Watson stat		1.630574
Prob(F-statistic)	0.000000

E13: Alterations and additions commencements (public, chain volume)

Dependent Variable: DLOG(CAAVOL_PUBLIC)
Method: Least Squares
Date: 12/21/18 Time: 14:15
Sample (adjusted): 1988Q3 2018Q2
Included observations: 120 after adjustments

Variable	Coefficient	Std. Error	t-Statistic	Prob.
C	0.329528	0.203860	1.616447	0.1087
LOG(CAAVOL_PUBLIC(−1))	−0.089483	0.055559	−1.610592	0.1100
DLOG(CAAVOL_PUBLIC(−1))	−0.409814	0.100261	−4.087488	0.0001
DLOG(CAAVOL_PUBLIC(−2))	−0.354218	0.098821	−3.584432	0.0005
DLOG(CAAVOL_PUBLIC(−3))	−0.067145	0.094554	−0.710124	0.4791
R-squared	0.245474	Mean dependent var		0.001079
Adjusted R-squared	0.219230	S.D. dependent var		0.442842
S.E. of regression	0.391301	Akaike info criterion		1.002094
Sum squared resid	17.60839	Schwarz criterion		1.118240
Log likelihood	−55.12565	Hannan-Quinn criter.		1.049261
F-statistic	9.353387	Durbin-Watson stat		2.010975
Prob(F-statistic)	0.000001

E14: Detached housing work done (chain volume)

Dependent Variable: DLOG(WDHOUSEVOL)
Method: Least Squares
Date: 12/21/18 Time: 14:15
Sample: 1988Q1 2018Q2
Included observations: 122

Variable	Coefficient	Std. Error	t-Statistic	Prob.
C	0.001749	0.002150	0.813601	0.4176
LOG(WDHOUSEVOL(−1))-LOG(CHOUSEVOL_PRIVATE(−1))	−0.470003	0.047843	−9.823940	0.0000
DLOG(WDHOUSEVOL(−1))	−0.009624	0.045602	−0.211036	0.8332
DLOG(WDHOUSEVOL(−2))	−0.080804	0.035412	−2.281850	0.0243
DLOG(CHOUSEVOL_PRIVATE)	0.423758	0.034183	12.39687	0.0000
GST	0.145296	0.024069	6.036560	0.0000
GST(−1)	−0.139260	0.033024	−4.216923	0.0000
R-squared	0.872137	Mean dependent var		0.005211
Adjusted R-squared	0.865466	S.D. dependent var		0.063272
S.E. of regression	0.023207	Akaike info criterion		−4.633037
Sum squared resid	0.061936	Schwarz criterion		−4.472150
Log likelihood	289.6152	Hannan-Quinn criter.		−4.567690
F-statistic	130.7335	Durbin-Watson stat		2.045568
Prob(F-statistic)	0.000000

E15: Higher-density housing work done (chain volume)

Dependent Variable: DLOG(WDOTHERVOL)
Method: Least Squares
Date: 12/21/18 Time: 14:15
Sample (adjusted): 1988Q3 2018Q2
Included observations: 120 after adjustments

Variable	Coefficient	Std. Error	t-Statistic	Prob.
C	0.006704	0.004129	1.623705	0.1072
LOG(WDOTHERVOL(−1))-LOG(COTHERVOL_PRIVATE(−1))	−0.194171	0.053496	−3.629636	0.0004
DLOG(COTHERVOL_PRIVATE)	0.130603	0.027947	4.673220	0.0000
DLOG(COTHERVOL_PRIVATE(−1))	0.037821	0.046647	0.810807	0.4192
DLOG(COTHERVOL_PRIVATE(−2))	0.073610	0.039601	1.858790	0.0657
DLOG(COTHERVOL_PRIVATE(−3))	0.078526	0.032522	2.414587	0.0174
GST	0.164214	0.045213	3.632027	0.0004
GST(−1)	−0.236190	0.045392	−5.203370	0.0000
R-squared	0.621852	Mean dependent var		0.017915
Adjusted R-squared	0.598218	S.D. dependent var		0.067410
S.E. of regression	0.042729	Akaike info criterion		−3.403547
Sum squared resid	0.204484	Schwarz criterion		−3.217714
Log likelihood	212.2128	Hannan-Quinn criter.		−3.328079
F-statistic	26.31148	Durbin-Watson stat		2.008478
Prob(F-statistic)	0.000000

E16: Alterations and additions work done (chain volume)

Dependent Variable: DLOG(WDAAVOL)
Method: Least Squares
Date: 12/21/18 Time: 14:15
Sample: 1988Q1 2018Q2
Included observations: 122

Variable	Coefficient	Std. Error	t-Statistic	Prob.
C	0.016480	0.002778	5.932315	0.0000
LOG(WDAAVOL(−1))-LOG(CAAVOL_PRIVATE(−1))	−0.546629	0.056386	−9.694322	0.0000
DLOG(CAAVOL_PRIVATE)	0.370454	0.042306	8.756560	0.0000
GST	0.101486	0.027391	3.705069	0.0003
GST(−1)	−0.244923	0.035605	−6.878841	0.0000
R-squared	0.797175	Mean dependent var		0.003186
Adjusted R-squared	0.790241	S.D. dependent var		0.058981
S.E. of regression	0.027013	Akaike info criterion		−4.344888
Sum squared resid	0.085374	Schwarz criterion		−4.229969
Log likelihood	270.0381	Hannan-Quinn criter.		−4.298211
F-statistic	114.9628	Durbin-Watson stat		2.220353
Prob(F-statistic)	0.000000

E17: Rental vacancy rate (%)

Dependent Variable: VACANCY/100
Method: Least Squares (Gauss-Newton / Marquardt steps)
Date: 12/21/18 Time: 14:15
Sample: 1983Q1 2018Q2
Included observations: 142
Convergence achieved after 6 iterations
White heteroskedasticity-consistent standard errors & covariance using outer product of gradients
VACANCY/100 = EXCESS_COMP + C(1)*(VACANCY(−1) – C(2))/100 + VACANCY(−1)/100 + C(3)*D(UR(−1)/100)

	Coefficient	Std. Error	t-Statistic	Prob.
C(1)	−0.151162	0.024864	−6.079593	0.0000
C(2)	2.412196	0.121069	19.92407	0.0000
C(3)	0.206798	0.082588	2.503981	0.0134
R-squared	0.919038	Mean dependent var		0.027989
Adjusted R-squared	0.917873	S.D. dependent var		0.007645
S.E. of regression	0.002191	Akaike info criterion		−9.388018
Sum squared resid	0.000667	Schwarz criterion		−9.325571
Log likelihood	669.5493	Hannan-Quinn criter.		−9.362642
Durbin-Watson stat	1.720493

E18: CPI rents (real, deflated using trimmed mean CPI)

Dependent Variable: DLOG(RENT_CPI)-INF_3Y
Method: Least Squares (Gauss-Newton / Marquardt steps)
Date: 12/21/18 Time: 14:15
Sample: 1983Q1 2018Q2
Included observations: 142
Convergence achieved after 4 iterations
White heteroskedasticity-consistent standard errors & covariance using outer product of gradients
DLOG(RENT_CPI)-INF_3Y = C(1)/400 + C(2)*(VACANCY(−5) – VACANCY_NATURAL)/100 + C(3)*D(VACANCY(−1)/100) + C(4) *D(VACANCY(−2)/100) + C(5)*D(VACANCY(−3)/100) + C(6) *D(VACANCY(−4)/100) + C(7)*(DLOG(RENT_CPI(−1))-INF_3Y(−1)-C(1)/400) + C(8)*(DLOG(RENT_CPI(−2))-INF_3Y(−2)-C(1)/400) + C(9) *((DLOG(RINC)+DLOG(RINC(−1)))/2-LR_INC) + C(10)*(DLOG(RINC(−2))-LR_INC)

	Coefficient	Std. Error	t-Statistic	Prob.
C(1)	0.901008	0.352870	2.553370	0.0118
C(2)	−0.102501	0.040655	−2.521242	0.0129
C(3)	−0.119479	0.097738	−1.222447	0.2237
C(4)	−0.262032	0.121775	−2.151767	0.0332
C(5)	−0.300454	0.097070	−3.095218	0.0024
C(6)	−0.175276	0.097626	−1.795394	0.0749
C(7)	0.529930	0.118757	4.462286	0.0000
C(8)	0.193711	0.107003	1.810343	0.0725
C(9)	0.036692	0.019010	1.930111	0.0557
C(10)	0.048468	0.012212	3.968832	0.0001
R-squared	0.800702	Mean dependent var		0.000355
Adjusted R-squared	0.787114	S.D. dependent var		0.005090
S.E. of regression	0.002349	Akaike info criterion		−9.202105
Sum squared resid	0.000728	Schwarz criterion		−8.993948
Log likelihood	663.3494	Hannan-Quinn criter.		−9.117518
Durbin-Watson stat	2.035132

E19: Housing prices (real, deflated using trimmed mean CPI)

Dependent Variable: DLOG(RDP*TCPI)-INF_3Y
Method: Least Squares
Date: 12/21/18 Time: 14:15
Sample: 1983Q1 2018Q2
Included observations: 142
White heteroskedasticity-consistent standard errors & covariance

Variable	Coefficient	Std. Error	t-Statistic	Prob.
C	0.142670	0.054624	2.611829	0.0100
LOG(RRENT(−1)/RDP(−1))-LOG(UC(−1))	0.022656	0.008718	2.598784	0.0104
DLOG(RDP(−1)*TCPI(−1))-INF_3Y(−1)	0.739314	0.067829	10.89963	0.0000
D(RVMR(−1))	−0.009176	0.002430	−3.776474	0.0002
D(RVMR(−2))	−0.005238	0.001890	−2.771782	0.0064
R-squared	0.692894	Mean dependent var		0.005482
Adjusted R-squared	0.683928	S.D. dependent var		0.020764
S.E. of regression	0.011673	Akaike info criterion		−6.028431
Sum squared resid	0.018669	Schwarz criterion		−5.924352
Log likelihood	433.0186	Hannan-Quinn criter.		−5.986137
F-statistic	77.27517	Durbin-Watson stat		1.596512
Prob(F-statistic)	0.000000	Wald F-statistic		62.04658
Prob(Wald F-statistic)	0.000000

E20: Adult population (15+ years)

Dependent Variable: DLOG(WAP)
Method: Least Squares
Date: 12/21/18 Time: 14:15
Sample: 1983Q1 2018Q2
Included observations: 142
White heteroskedasticity-consistent standard errors & covariance

Variable	Coefficient	Std. Error	t-Statistic	Prob.
C	0.000287	0.000136	2.106600	0.0369
DLOG(WAP(−1))	0.514310	0.078829	6.524362	0.0000
DLOG(WAP(−2))	0.414611	0.078701	5.268156	0.0000
R-squared	0.809737	Mean dependent var		0.004009
Adjusted R-squared	0.806999	S.D. dependent var		0.000811
S.E. of regression	0.000356	Akaike info criterion		−13.01993
Sum squared resid	1.77E-05	Schwarz criterion		−12.95748
Log likelihood	927.4147	Hannan-Quinn criter.		−12.99455
F-statistic	295.7839	Durbin-Watson stat		1.879626
Prob(F-statistic)	0.000000	Wald F-statistic		367.9006
Prob(Wald F-statistic)	0.000000

E21: Real household disposable income

Dependent Variable: DLOG(RINC_PER_CAPITA)
Method: Least Squares (Gauss-Newton / Marquardt steps)
Date: 12/21/18 Time: 14:15
Sample (adjusted): 1985Q2 2018Q2
Included observations: 133 after adjustments
White heteroskedasticity-consistent standard errors & covariance
DLOG(RINC_PER_CAPITA) = C(1) + C(2)*DLOG(RINC_PER_CAPITA(−1)) + C(3)*DLOG(RINC_PER_CAPITA(−2)) + C(4)*DLOG(RINC_PER_CAPIT A(−3)) + C(5)*DLOG(RINC_PER_CAPITA(−4)) + C(6)*GST(−1) – 0.0016 *((RVMR(−1)-RVMR(−41)))

	Coefficient	Std. Error	t-Statistic	Prob.
C(1)	0.003605	0.001225	2.943859	0.0039
C(2)	−0.207404	0.115951	−1.788723	0.0760
C(3)	0.091408	0.096884	0.943483	0.3472
C(4)	0.158653	0.094180	1.684576	0.0945
C(5)	0.049565	0.079508	0.623391	0.5341
C(6)	0.058458	0.002246	26.03123	0.0000
R-squared	0.068745	Mean dependent var		0.002491
Adjusted R-squared	0.032081	S.D. dependent var		0.014902
S.E. of regression	0.014661	Akaike info criterion		−5.563235
Sum squared resid	0.027297	Schwarz criterion		−5.432843
Log likelihood	375.9551	Hannan-Quinn criter.		−5.510249
F-statistic	1.875019	Durbin-Watson stat		1.451208
Prob(F-statistic)	0.103194	Wald F-statistic		573.6565
Prob(Wald F-statistic)	0.000000

E.2 Identities and Calibrated Equations Used for Simulations

E22

c a s h_{t} = c a s h_e x p_{t}

E23

v m r_{t} = v m r_{t - 1} + Δ (c a s h_{t})

E24

r v m r_{t} = v m r_{t} - ({(\frac{t c p i}{t c p i_{t - 12}})}^{\frac{1}{3}} - 1) * 100

E25

i_10 y_b o n d_{t} = \exp (\frac{1}{40} \sum_{i = 1}^{40} \ln (c a s h_e x p_{t + i}))

E26

\begin{array}{l} u c_{t} = r v m r_e x p_{t} + u c_r u n n i n g_c o s t_{t} + u c_t r a n s a c t i o n_c o s t_{t} + u c_d e p r e c i a t i o n_{t} \\ - u c_a p p r e c i a t i o n_{t} \end{array}

E27

r v m r_e x p_{t} = (1 + \frac{v m r_e x p_{t}}{100}) / (1 + \frac{i n f_e x p_{t}}{100}) * 100 - 100

E28

v m r_e x p_{t} = v m r + (i_10 y_b o n d_{t} - c a s h_{t}) - a v e r a g e_s p r e a d_{t}

E29

i n f_e x p_{t} = 0.9 * i n f_e x p_{t - 1} + 0.1 * (\frac{t c p i_{t}}{t c p i_{t - 4}} - 1) * 100

E30

u c_a p p r e c i a t i o n_{t} = u c_a p p r e c i a t i o n_{t - 1}

E31

u c_d e p r e c i a t i o n_{t} = u c_d e p r e c i a t i o n_{t - 1}

E32

u n_t r a n s a c t i o n_c o s t_{t} = u c_t r a n s a c t i o n_c o s t_{t - 1}

E33

u n_r u n n i n g_c o s t_{t} = u c_r u n n i n g_c o s t_{t - 1}

E34

\begin{array}{l} Δ \ln (t c p i_{t}) = 0.30 * Δ \ln (t c p i_{t - 1}) + 0.25 * Δ \ln (t c p i_{t - 2}) + 0.20 * Δ \ln (t c p i_{t - 3}) \\ + 0.15 * Δ \ln (t c p i_{t - 4}) + 0.1 * 0.025 / 4 \end{array}

E35

i n f_3 y_{t} = (\ln (t c p i_{t}) - \ln (t c p i_{t - 12})) / 12

E36

r r e n t_{t} = r e n t_c p i_{t} / t c p i_{t}

E37

y i e l d_{t} = r r e n t_{t} / r d p_{t} * y i e l d_a d j * 100

E38

c o m p t o t a l n o_{t} = c o m p h o u s e n o_{t} + c o m p o t h e r n o_{t}

E39

s t o c k_n o_{t} = s t o c k_n o_{t - 1} + c o m p t o t a l n o_{t} * (1 - d e m o l i t i o n_r a t e_{t})

E40

s t o c k_{t} = s t o c k_{t - 1} * (1 - d e p r e c i a t i o n_r a t e_{t}) + n a t d i

E41

d e m o l i t i o n_r a t e_{t} = \frac{1}{20} \sum_{i = 1}^{20} d e m o l i t i o n_r a t e_{t - 1}

E42

d e p r e c i a t i o n_r a t e_{t} = \frac{1}{4} \sum_{i = 1}^{4} d e p r e c i a t i o n_r a t e_{t - 1}

E43

\begin{array}{l} Δ \ln (b a h o u s e a v o l_{t}) = λ_{h o u s e} * (\log (\frac{b a h o u s e a v o l_{t - 1}}{(\frac{r i n c_{t - 1}}{w a p_{t - 1}})}) - \frac{1}{8} \sum_{i = 1}^{8} \log (\frac{b a h o u s e a v o l_{t - i}}{(\frac{r i n c_{t - i}}{w a p_{t - i}})})) \\ + l r_i n c_p e r_w a p \end{array}

E44

b a h o u s e n o_{t} = b a h o u s e v o l_{t} / b a h o u s e a v o l_{t} * 1000

E45

\begin{array}{l} Δ \ln (b a o t h e r a v o l_{t}) = λ_{o t h e r} * (\log (\frac{b a o t h e r a v o l_{t - 1}}{(\frac{r i n c_{t - 1}}{w a p_{t - 1}})}) - \frac{1}{8} \sum_{i = 1}^{8} \log (\frac{b a o t h e r a v o l_{t - i}}{(\frac{r i n c_{t - i}}{w a p_{t - i}})})) \\ + l r_i n c_p e r_w a p \end{array}

E46

b a o t h e r n o_{t} = b a o t h e r v o l_{t} / b a o t h e r a v o l_{t} * 1000

E47

c h o u s e v o l_p r i v a t e_{t} = c h o u s e v o l_{t} - c h o u s e v o l_p u b l i c_{t}

E48

Δ \ln (n a h o u s e v o l_{t}) = Δ \ln (w d h o u s e v o l_{t})

E49

c o t h e r v o l_p r i v a t e_{t} = c o t h e r v o l_{t} - c o t h e r v o l_p u b l i c_{t}

E50

Δ \ln (n a o t h e r v o l_{t}) = Δ \ln (w d o t h e r v o l_{t})

E51

c a a v o l_p r i v a t e_{t} = c a a v o l_{t} - c a a v o l_p u b l i c_{t}

E52

Δ \ln (n a a a v o l_{t}) = Δ \ln (w d a a v o l_{t})

E53

b a t o t a l v o l_{t} = b a a a v o l_{t} + b a h o u s e v o l_{t} + b a o t h e r v o l_{t}

E54

b a t o t a l n o_{t} = b a h o u s e n o_{t} + b a o t h e r n o_{t}

E55

n a n e w v o l_{t} = n a h o u s e v o l_{t} + n a o t h e r v o l_{t}

E56

n a t d i_{t} = n a n e w v o l_{t} + n a a a v o l_{t}

E57

n a t d i v a l_{t} = n a t d i_{t} * i p d_n a t d i_{t} / 100

E58

i n v_t o_i n c o m e_{t} = n a t d i_{t} / r i n c_{t}

E59

r i n c_{t} = r i n c_p e r_c a p i t a_{t} * w a p_{t}

E60

a h s_{t} = \frac{w a p_{t}}{s t o c k_n o_{t}} * 1000

E61

e x c e s s_c o m p_{t} = (Δ (s t o c k_n o_{t}) - \frac{Δ (w a p_{t})}{(\frac{1}{20} \sum_{i = 1}^{20} a h s_{t - i + 1})}) / s t o c k_n o_{t}

E62

Δ (u r_{t}) = - 0.5 * (Δ \ln (r i n c_{t}) - l r_i n c) * 100