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126
7 Appendices
Appendix A ... 2
Appendix B ... 2
Appendix C ... 3
Appendix D ... 4
Appendix E ... 5
Appendix F ... 6
Appendix G ... 6
Appendix H ... 7
Appendix I ... 8
Appendix J... 8
Appendix K ... 9
Appendix L ... 9
Appendix M ... 10
Appendix N ... 10
Appendix O ... 10
Appendix P ... 11
Appendix Q ... 11
Appendix R ... 12
Appendix S ... 12
2
Figure 19: Disposable income 1973Q1-2017Q2 (Danmark’s Nationalbank)
Appendix B
Figure 20 Change in housing prices and user cost 1974Q4-2017Q2 (Danmark’s Nationalbank and own estimations)
3 Estimation variables overview
𝐾𝑑= 𝑑𝑒𝑚𝑎𝑛𝑑
𝑃𝐶𝑃 = 𝑝𝑟𝑖𝑐𝑒 𝑜𝑓 𝑝𝑟𝑖𝑣𝑎𝑡𝑒 𝑐𝑜𝑛𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛, 𝑑𝑒𝑓𝑙𝑎𝑡𝑜𝑟 𝑓𝐶𝑝𝑢𝑥ℎ = 𝑐𝑜𝑛𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛 𝑒𝑥𝑐𝑙. ℎ𝑜𝑢𝑠𝑖𝑛𝑔
𝑢 = 𝑢𝑠𝑒𝑟 𝑐𝑜𝑠𝑡
𝑦 = 𝑓𝑖𝑟𝑠𝑡 − 𝑦𝑒𝑎𝑟 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 𝐴𝐼𝐻 = ℎ𝑜𝑢𝑠𝑖𝑛𝑔 𝑠𝑡𝑜𝑐𝑘
𝐴𝐼𝐻𝑤 = ℎ𝑜𝑢𝑠𝑖𝑛𝑔 𝑠𝑡𝑜𝑐𝑘 𝑤𝑎𝑛𝑡𝑒𝑑 𝜀 = 𝑒𝑟𝑟𝑜𝑟 𝑡𝑒𝑟𝑚/𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙𝑠
𝑐𝑡= 𝑢𝑡𝑖𝑙𝑖𝑡𝑦 𝑓𝑟𝑜𝑚 𝑐𝑜𝑛𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 𝑔𝑜𝑜𝑑 𝐼 = 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
𝑃0= 𝑛𝑒𝑡 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑟𝑖𝑐𝑒 Π = 𝑝𝑟𝑜𝑓𝑖𝑡
𝑖𝑡= 𝑠ℎ𝑜𝑟𝑡 − 𝑡𝑒𝑟𝑚 𝑛𝑜𝑚𝑖𝑛𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑟∗ = 𝑛𝑒𝑢𝑡𝑟𝑎𝑙 𝑟𝑒𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒
𝑝𝑡 = 𝑎𝑐𝑡𝑢𝑎𝑙 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝑝∗ = 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝑡𝑎𝑟𝑔𝑒𝑡 𝑌 = 𝑎𝑐𝑡𝑢𝑎𝑙 𝑜𝑢𝑡𝑝𝑢𝑡 𝑌∗ = 𝑜𝑢𝑡𝑝𝑢𝑡 𝑡𝑎𝑟𝑔𝑒𝑡 𝑂𝑃 = 𝑜𝑢𝑡𝑝𝑢𝑡 𝑔𝑎𝑝
𝑀𝑃𝑂𝐿𝑅𝐸𝑁𝑇 = 𝑎𝑐𝑡𝑢𝑎𝑙 𝑜𝑓𝑓𝑖𝑐𝑖𝑎𝑙 𝑏𝑎𝑛𝑘 𝑟𝑎𝑡𝑒
4
Figure 21: Inflation rate Denmark 1973Q1-2017Q2 (Danmark’s Nationalbank)
5
Many time series are trending, and it is important to distinguish between deterministic (time) trend and stochastic trend (Nielsen, 2005). The equations for the two types of trends are presented in the following:
𝑆𝑡𝑜𝑐ℎ𝑎𝑠𝑡𝑖𝑐 𝑡𝑟𝑒𝑛𝑑: ∆𝑌𝑡= 𝛼 + 𝛿𝑌𝑡−1+ 𝑢𝑡
𝑆𝑡𝑜𝑐ℎ𝑎𝑠𝑡𝑖𝑐 𝑎𝑛𝑑 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑖𝑠𝑡𝑖𝑐 𝑡𝑟𝑒𝑛𝑑: ∆𝑌𝑡= 𝛼 + 𝛽𝑡 + 𝛿𝑌𝑡−1+ 𝑢𝑡
(Stadnytska, 2010).
where
𝛼 is a constant 𝛿 is the autoregression parameter
𝑢𝑡 is the non-systematic component 𝛽𝑡 is the deterministic time trend variable
The distinction between the two types of trend is vital for the understanding of long-term behaviour of time series. To determine whether or not the time series was exhibiting a time trend, the term “trend” was applied when conducting the unit root test in R. The time trend coefficient was kept if it came out significant, indicating that the time series had a time trend. If the coefficient was not significant, it was disregarded and the unit root test was instead performed using “drift”.
6 Figure 22: Illustration of a unit root (authors’ own creation).
Appendix G
Figure 23: Variance Inflation Factor test results: Housing price model without synthetic interest rates (authors’ own creation).
Variable Name Result
Lagged change in real housing price Dlog(KP/PCP) 1.107806
First-year payment RENTEMIN+SSATS+AFDR 1.475738
Change in interest and property taxation rate D(RENTE30+SSATS)*DB98K4 1.467093
User cost RENTE30+SSATS-DPCPE-DRKPE 1.107514
Disposable income relative to housing stock log(YD_H/PCP)-log(AIH) 1.031336 Variance Inflation Factor (VIF) - Actual Housing Price Model
7
Figure 24: Cook’s D plot: Housing price model without synthetic interest rates (authors’ own creation).
8
Figure 25: Durbin-Watson test results: Housing price model without synthetic interest rates (authors’ own creation).
Appendix J
Figure 26: Granger causality test results: Housing price model without synthetic interest rates (authors’ own creation).
Lag Autocorrelation D-W Statistic p-value
1 -0.024867 2.045156 0.934
2 0.060122 1.866853 0.336
3 0.108356 1.768210 0.134
4 -0.032429 2.044869 0.624
5 0.119163 1.732171 0.134
Durbin-Watson - Actual Housing Price Model
First-year payment
Model 1: KP - Lags(KP, 1:1) + Lags(fypayment, 1:1) Model 2: KP - Lags(KP, 1:1)
Res.Df Df F-value p-value
167
168 -1 0.1478 0.7011
Change in interest and property taxation rate Model 1: KP - Lags(KP, 1:1) + Lags(inttax, 1:1) Model 2: KP - Lags(KP, 1:1)
Res.Df Df F-value p-value
167
168 -1 0.1019 0.7499
User cost
Model 1: KP - Lags(KP, 1:4) + Lags(usercost, 1:4) Model 2: KP - Lags(KP, 1:4)
Res.Df Df F-value p-value
158
162 -4 0.5128 0.7264
Disposable income relative to housing stock
Model 1: KP - Lags(KP, 1:6) + Lags(realincstock, 1:6) Model 2: KP - Lags(KP, 1:6)
Res.Df Df F-value p-value
152
158 -6 0.6467 0.6927
Granger Causality Test - Actual Housing Price Model
9
Figure 27: Table of best test results for the Taylor rate regression with estimated coefficients in the full estimation period (authors’ own creation). Variable for inflation is not lagged any periods back. The Newey-West robust standard errors are given in the “Std.Error” column.
Appendix L
Figure 28: The actual official bank rate and the estimated insignificant and unlagged Taylor rate for the full period (Statistics Denmark and authors’ own creation).
Variable Name Coefficient Std.Error t-value p-value
Inflation gap DlogPCP 1.93491 0.39692 4.8748 <2.442E-06***
Output gap OP -0.20603 0.22362 -0.9213 0.3582
Constant 0.07367 0.00466 15.8105 <2.2E-16 ***
Signif. Codes: 0.001 '***' 0.01 '**' 0.05 '*' 0.1 '.'
Estimation period: 1973Q1-2017Q2 R-squared: 0.3241 Adjusted R-squared: 0.3160 Estimated Taylor Rate
10
Figure 29: Table of test results for the Taylor rate regression with estimated coefficients in the full estimation period (authors’ own creation). Inflation is lagged four quarters back in time.
Appendix N
Figure 30: Durbin-Watson test results: Taylor rate (authors’own creation).
Appendix O
Figure 31: Variance Inflation Factor test results: Taylor rate (authors’ own creation).
Variable Name Coefficient Std.Error t-value p-value
Inflation gap DlogPCP- 4 1.88604 0.45831 4.1152 <6.034E-05***
Output gap OP -0.11425 0.24113 -0.4738 0.6362
Constant 0.07197 0.00468 15.3763 <2.2E-16 ***
Signif. Codes: 0.001 '***' 0.01 '**' 0.05 '*' 0.1 '.'
Estimation period: 1973Q1-2017Q2 R-squared: 0.3344 Adjusted R-squared: 0.3262 Estimated Taylor Rate
Lag Autocorrelation D-W Statistic p-value
1 0.923428 0.109882 0
2 0.834597 0.247033 0
3 0.749016 0.394424 0
4 0.647926 0.566896 0
5 0.567644 0.793171 0
Durbin-Watson - Taylor Rate
Inflation
Model 1: MPOLRENT - Lags(MPOLRENT, 1:4) + Lags(logPCP, 1:4) Model 2: MPOLRENT - Lags(MPOLRENT, 1:4)
Res.Df Df F-value p-value
159
163 -4 2.4155 0.051 .
Output gap
Model 1: MPOLRENT - Lags(MPOLRENT, 1:1) + Lags(OP, 1:1) Model 2: MPOLRENT - Lags(MPOLRENT, 1:1)
Res.Df Df F-value p-value
168
169 -1 6.4748 0.01184 *
Granger Causality Test - Taylor Rate
11
Figure 32: Variance Inflation Factor test results: Modified housing price model with synthetic interest rates (authors’ own creation).
Appendix Q
Figure 33: Cook’s D plot: Modified housing price model with synthetic interest rates (authors’ own creation).
Variable Name Result
Lagged change in real housing price Dlog(KP/PCP) 1.079103
First-year payment RENTEMIN+SSATS+AFDR 1.142134
Change in interest and property taxation rate D(RENTE30+SSATS)*DB98K4 1.153672
User cost RENTE30+SSATS-DPCPE-DRKPE 1.045852
Disposable income relative to housing stock log(YD_H/PCP)-log(AIH) 1.031954 Variance Inflation Factor (VIF) - Modified Housing Price Model
12
Figure 34: Durbin-Watson test results: Modified housing price model with synthetic interest rates (authors’
own creation).
Appendix S
Figure 35: Granger causality test results: Modified housing price model with synthetic interest rates (authors’
own creation).
Lag Autocorrelation D-W Statistic p-value
1 -0.027121 2.008941 0.990
2 0.096603 1.758905 0.084
3 0.065727 1.815627 0.246
4 -0.209616 2.364342 0.006
5 0.128306 1.664306 0.042
Durbin-Watson - Modified Housing Price Model
First-year payment
Model 1: KP - Lags(KP, 1:1) + Lags(fypayment, 1:1) Model 2: KP - Lags(KP, 1:1)
Res.Df Df F-value p-value
167
168 -1 0.3586 0.5501
Change in interest and property taxation rate Model 1: KP - Lags(KP, 1:1) + Lags(inttax, 1:1) Model 2: KP - Lags(KP, 1:1)
Res.Df Df F-value p-value
167
168 -1 1.8057 0.1808
User cost
Model 1: KP - Lags(KP, 1:3) + Lags(usercost, 1:3) Model 2: KP - Lags(KP, 1:3)
Res.Df Df F-value p-value
161
164 -3 3.5555 0.01574
Disposable income relative to housing stock
Model 1: KP - Lags(KP, 1:6) + Lags(realincstock, 1:6) Model 2: KP - Lags(KP, 1:6)
Res.Df Df F-value p-value
152
158 -6 0.6467 0.6927
Granger Causality Test - Modified Housing Price Model