Financial Time Series Models
Realty Income Corporation (O) is revered as one of the most reliable dividend-paying stocks in the market today. For nearly 30 years as a publicly traded real estate investment trust (REIT), Realty Income has provided investors with outsized returns compared to the S&P 500 while paying an attractive dividend yield and consistently raising its dividend payouts. For the I will used Financial time series models to forecast the stock prices and evaluate if the trend follows home ownership rate in the us. The Realty Income Corporation (O) is a real estate investment trust (REIT) that primarily invests in retail and commercial properties, such as shopping centers, drug stores, and convenience stores. While it is not directly related to home ownership rate, it can have an indirect impact on the real estate market and thus the home ownership rate. As a REIT, Realty Income Corporation invests in real estate properties and generates income from rent payments from its tenants. The success or failure of Realty Income Corporation can have an impact on the overall real estate market, including supply and demand for commercial real estate properties, which in turn can affect the prices of residential real estate properties and the home ownership rate. Additionally, the performance of Realty Income Corporation can reflect the overall health of the real estate market, which can impact home ownership rates.
Figure 1: Realty Income Corporation (O) Stock Prices Throughout Time
Model Fitting
Look at the ACF, PACF plots of the returns
Figure 2.
ADF Test
data: returns
Dickey-Fuller = -19.727, Lag order = 19, p-value = 0.01
alternative hypothesis: stationaryBased on the ACF and the ADF.test we can see that the data is stationary.
Acf
data: returns
Chi-squared = 323.74, df = 1, p-value < 2.2e-16Because the p-value is < 0.05, we reject the null hypothesis and conclude the presence of ARCH(1) effects.
p d q AIC BIC AICc
1 0 0 0 21944.245 21957.986 21944.246
2 0 1 0 -37048.962 -37042.091 -37048.961
3 0 2 0 -31210.805 -31203.935 -31210.805
4 0 0 1 12282.056 12302.669 12282.060
5 0 1 1 -37176.467 -37162.725 -37176.465
6 0 2 1 -37037.957 -37024.215 -37037.955
7 0 0 2 3648.326 3675.809 3648.331
8 0 1 2 -37174.737 -37154.125 -37174.733
9 0 2 2 -37167.371 -37146.760 -37167.368
10 1 0 0 -37041.126 -37020.514 -37041.123
11 1 1 0 -37176.957 -37163.216 -37176.956
12 1 2 0 -33955.641 -33941.900 -33955.639
13 1 0 1 -37168.068 -37140.585 -37168.062
14 1 1 1 -37175.015 -37154.403 -37175.012
15 1 2 1 -37167.592 -37146.980 -37167.589
16 1 0 2 -37166.811 -37132.457 -37166.802
17 1 1 2 -37173.173 -37145.691 -37173.168
18 1 2 2 -37165.760 -37138.278 -37165.755
19 2 0 0 -37169.046 -37141.563 -37169.040
20 2 1 0 -37175.032 -37154.420 -37175.028
21 2 2 0 -34887.046 -34866.434 -34887.042
22 2 0 1 -37166.806 -37132.452 -37166.798
23 2 1 1 -37173.131 -37145.649 -37173.126
24 2 2 1 -37165.719 -37138.237 -37165.713
25 2 0 2 -37164.872 -37123.648 -37164.861
26 2 1 2 -37197.328 -37162.974 -37197.319
27 2 2 2 -37163.645 -37129.293 -37163.637
28 NA NA NA NA NA NA
29 NA NA NA NA NA NA
30 NA NA NA NA NA NA
31 NA NA NA NA NA NA
32 NA NA NA NA NA NA
33 NA NA NA NA NA NA
34 NA NA NA NA NA NA
35 NA NA NA NA NA NA
36 NA NA NA NA NA NA
37 NA NA NA NA NA NA
38 NA NA NA NA NA NA
39 NA NA NA NA NA NA
40 NA NA NA NA NA NA
41 NA NA NA NA NA NA
42 NA NA NA NA NA NA
43 NA NA NA NA NA NA
44 NA NA NA NA NA NA
45 NA NA NA NA NA NA
46 NA NA NA NA NA NA
47 NA NA NA NA NA NA
48 NA NA NA NA NA NA
49 NA NA NA NA NA NA
50 NA NA NA NA NA NA p d q AIC BIC AICc
26 2 1 2 -37197.33 -37162.97 -37197.32 p d q AIC BIC AICc
11 1 1 0 -37176.96 -37163.22 -37176.96 p d q AIC BIC AICc
26 2 1 2 -37197.33 -37162.97 -37197.32Auto Arima
Series: log(ots)
ARIMA(1,1,0) with drift
Coefficients:
ar1 drift
-0.1355 5e-04
s.e. 0.0117 2e-04
sigma^2 = 0.0003157: log likelihood = 18594.47
AIC=-37182.94 AICc=-37182.94 BIC=-37162.33Auto arima suggests ARIMA(1,1,0)
initial value -4.021303
iter 2 value -4.030568
iter 2 value -4.030568
iter 2 value -4.030568
final value -4.030568
converged
initial value -4.030521
iter 1 value -4.030521
final value -4.030521
convergedFigure.
$fit
Call:
arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S),
xreg = constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc,
REPORT = 1, reltol = tol))
Coefficients:
ar1 constant
-0.1355 5e-04
s.e. 0.0117 2e-04
sigma^2 estimated as 0.0003156: log likelihood = 18594.47, aic = -37182.94
$degrees_of_freedom
[1] 7118
$ttable
Estimate SE t.value p.value
ar1 -0.1355 0.0117 -11.5369 0.0000
constant 0.0005 0.0002 2.8192 0.0048
$AIC
[1] -5.222323
$AICc
[1] -5.222323
$BIC
[1] -5.219428 initial value -4.021244
iter 2 value -4.021317
iter 3 value -4.030509
iter 4 value -4.030525
iter 5 value -4.030541
iter 6 value -4.030563
iter 7 value -4.030644
iter 8 value -4.030820
iter 9 value -4.031128
iter 10 value -4.031389
iter 11 value -4.031602
iter 12 value -4.031627
iter 13 value -4.031632
iter 14 value -4.031688
iter 15 value -4.031931
iter 16 value -4.032092
iter 17 value -4.032238
iter 18 value -4.032249
iter 19 value -4.032359
iter 20 value -4.032370
iter 21 value -4.032402
iter 22 value -4.032482
iter 23 value -4.032558
iter 24 value -4.032560
iter 25 value -4.032563
iter 26 value -4.032567
iter 27 value -4.032591
iter 28 value -4.032591
iter 28 value -4.032591
iter 28 value -4.032591
final value -4.032591
converged
initial value -4.032621
iter 1 value -4.032621
final value -4.032621
converged
$fit
Call:
arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S),
xreg = constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc,
REPORT = 1, reltol = tol))
Coefficients:
ar1 ar2 ma1 ma2 constant
0.4445 0.2376 -0.5878 -0.1810 5e-04
s.e. 0.1295 0.0684 0.1295 0.0789 2e-04
sigma^2 estimated as 0.0003143: log likelihood = 18609.42, aic = -37206.83
$degrees_of_freedom
[1] 7115
$ttable
Estimate SE t.value p.value
ar1 0.4445 0.1295 3.4338 0.0006
ar2 0.2376 0.0684 3.4750 0.0005
ma1 -0.5878 0.1295 -4.5383 0.0000
ma2 -0.1810 0.0789 -2.2941 0.0218
constant 0.0005 0.0002 3.4070 0.0007
$AIC
[1] -5.225679
$AICc
[1] -5.225678
$BIC
[1] -5.219889Fit Using ARIMA(2,1,2)
summary(garchFit(~garch(1,1), arima.res,trace = F))
Title:
GARCH Modelling
Call:
garchFit(formula = ~garch(1, 1), data = arima.res, trace = F)
Mean and Variance Equation:
data ~ garch(1, 1)
<environment: 0x140c47948>
[data = arima.res]
Conditional Distribution:
norm
Coefficient(s):
mu omega alpha1 beta1
3.4484e-05 7.2658e-06 1.2435e-01 8.4479e-01
Std. Errors:
based on Hessian
Error Analysis:
Estimate Std. Error t value Pr(>|t|)
mu 3.448e-05 1.420e-04 0.243 0.808
omega 7.266e-06 9.625e-07 7.549 4.37e-14 ***
alpha1 1.243e-01 1.003e-02 12.396 < 2e-16 ***
beta1 8.448e-01 1.230e-02 68.666 < 2e-16 ***
---
Signif. codes: 0 39;***39; 0.001 39;**39; 0.01 39;*39; 0.05 39;.39; 0.1 39; 39; 1
Log Likelihood:
20631.71 normalized: 2.897305
Description:
Mon Apr 17 21:04:56 2023 by user:
Standardised Residuals Tests:
Statistic p-Value
Jarque-Bera Test R Chi^2 1519.646 0
Shapiro-Wilk Test R W NA NA
Ljung-Box Test R Q(10) 54.80962 3.427468e-08
Ljung-Box Test R Q(15) 59.74675 2.787883e-07
Ljung-Box Test R Q(20) 62.13639 3.309501e-06
Ljung-Box Test R^2 Q(10) 10.36577 0.4090097
Ljung-Box Test R^2 Q(15) 31.42486 0.007703236
Ljung-Box Test R^2 Q(20) 32.75262 0.03590994
LM Arch Test R TR^2 26.89586 0.007998371
Information Criterion Statistics:
AIC BIC SIC HQIC
-5.793487 -5.789627 -5.793487 -5.792158 Series: data
ARIMA(2,1,2) with drift
Coefficients:
ar1 ar2 ma1 ma2 drift
0.4445 0.2376 -0.5878 -0.1810 5e-04
s.e. 0.1295 0.0684 0.1295 0.0789 2e-04
sigma^2 = 0.0003145: log likelihood = 18609.42
AIC=-37206.83 AICc=-37206.82 BIC=-37165.61
Training set error measures:
ME RMSE MAE MPE MAPE MASE
Training set 3.448358e-06 0.01772652 0.01111665 0.00255547 0.554396 0.04972213
ACF1
Training set 0.002687182
Title:
GARCH Modelling
Call:
garchFit(formula = ~garch(1, 1), data = arima.res, trace = F)
Mean and Variance Equation:
data ~ garch(1, 1)
<environment: 0x134af7ea8>
[data = arima.res]
Conditional Distribution:
norm
Coefficient(s):
mu omega alpha1 beta1
3.4484e-05 7.2658e-06 1.2435e-01 8.4479e-01
Std. Errors:
based on Hessian
Error Analysis:
Estimate Std. Error t value Pr(>|t|)
mu 3.448e-05 1.420e-04 0.243 0.808
omega 7.266e-06 9.625e-07 7.549 4.37e-14 ***
alpha1 1.243e-01 1.003e-02 12.396 < 2e-16 ***
beta1 8.448e-01 1.230e-02 68.666 < 2e-16 ***
---
Signif. codes: 0 39;***39; 0.001 39;**39; 0.01 39;*39; 0.05 39;.39; 0.1 39; 39; 1
Log Likelihood:
20631.71 normalized: 2.897305
Description:
Mon Apr 17 21:04:56 2023 by user:
Standardised Residuals Tests:
Statistic p-Value
Jarque-Bera Test R Chi^2 1519.646 0
Shapiro-Wilk Test R W NA NA
Ljung-Box Test R Q(10) 54.80962 3.427468e-08
Ljung-Box Test R Q(15) 59.74675 2.787883e-07
Ljung-Box Test R Q(20) 62.13639 3.309501e-06
Ljung-Box Test R^2 Q(10) 10.36577 0.4090097
Ljung-Box Test R^2 Q(15) 31.42486 0.007703236
Ljung-Box Test R^2 Q(20) 32.75262 0.03590994
LM Arch Test R TR^2 26.89586 0.007998371
Information Criterion Statistics:
AIC BIC SIC HQIC
-5.793487 -5.789627 -5.793487 -5.792158
meanForecast meanError standardDeviation lowerInterval upperInterval
1 3.448358e-05 0.01141431 0.01141431 -0.02233716 0.02240613
2 3.448358e-05 0.01155557 0.01155557 -0.02261402 0.02268299
3 3.448358e-05 0.01169084 0.01169084 -0.02287914 0.02294811
4 3.448358e-05 0.01182046 0.01182046 -0.02313318 0.02320215
5 3.448358e-05 0.01194473 0.01194473 -0.02337676 0.02344572
6 3.448358e-05 0.01206395 0.01206395 -0.02361042 0.02367939
7 3.448358e-05 0.01217837 0.01217837 -0.02383468 0.02390365
8 3.448358e-05 0.01228825 0.01228825 -0.02405004 0.02411900
9 3.448358e-05 0.01239380 0.01239380 -0.02425692 0.02432589
10 3.448358e-05 0.01249525 0.01249525 -0.02445575 0.02452472
11 3.448358e-05 0.01259278 0.01259278 -0.02464692 0.02471589
12 3.448358e-05 0.01268659 0.01268659 -0.02483078 0.02489975
13 3.448358e-05 0.01277685 0.01277685 -0.02500768 0.02507665
14 3.448358e-05 0.01286372 0.01286372 -0.02517794 0.02524691
15 3.448358e-05 0.01294735 0.01294735 -0.02534185 0.02541082
16 3.448358e-05 0.01302788 0.01302788 -0.02549970 0.02556867
17 3.448358e-05 0.01310546 0.01310546 -0.02565175 0.02572072
18 3.448358e-05 0.01318021 0.01318021 -0.02579826 0.02586722
19 3.448358e-05 0.01325225 0.01325225 -0.02593945 0.02600842
20 3.448358e-05 0.01332169 0.01332169 -0.02607556 0.02614453
21 3.448358e-05 0.01338865 0.01338865 -0.02620679 0.02627576
22 3.448358e-05 0.01345322 0.01345322 -0.02633335 0.02640232
23 3.448358e-05 0.01351551 0.01351551 -0.02645543 0.02652440
24 3.448358e-05 0.01357560 0.01357560 -0.02657320 0.02664217
25 3.448358e-05 0.01363358 0.01363358 -0.02668685 0.02675581
26 3.448358e-05 0.01368954 0.01368954 -0.02679652 0.02686549
27 3.448358e-05 0.01374356 0.01374356 -0.02690239 0.02697136
28 3.448358e-05 0.01379570 0.01379570 -0.02700459 0.02707356
29 3.448358e-05 0.01384605 0.01384605 -0.02710328 0.02717224
30 3.448358e-05 0.01389467 0.01389467 -0.02719857 0.02726754
31 3.448358e-05 0.01394163 0.01394163 -0.02729061 0.02735957
32 3.448358e-05 0.01398699 0.01398699 -0.02737951 0.02744847
33 3.448358e-05 0.01403081 0.01403081 -0.02746539 0.02753436
34 3.448358e-05 0.01407314 0.01407314 -0.02754837 0.02761733
35 3.448358e-05 0.01411405 0.01411405 -0.02762855 0.02769751
36 3.448358e-05 0.01415358 0.01415358 -0.02770603 0.02777500
37 3.448358e-05 0.01419179 0.01419179 -0.02778091 0.02784988
38 3.448358e-05 0.01422872 0.01422872 -0.02785330 0.02792226
39 3.448358e-05 0.01426442 0.01426442 -0.02792326 0.02799223
40 3.448358e-05 0.01429893 0.01429893 -0.02799091 0.02805988
41 3.448358e-05 0.01433230 0.01433230 -0.02805631 0.02812527
42 3.448358e-05 0.01436456 0.01436456 -0.02811954 0.02818851
43 3.448358e-05 0.01439576 0.01439576 -0.02818069 0.02824966
44 3.448358e-05 0.01442594 0.01442594 -0.02823983 0.02830880
45 3.448358e-05 0.01445512 0.01445512 -0.02829702 0.02836599
46 3.448358e-05 0.01448334 0.01448334 -0.02835234 0.02842131
47 3.448358e-05 0.01451064 0.01451064 -0.02840585 0.02847482
48 3.448358e-05 0.01453705 0.01453705 -0.02845761 0.02852658
49 3.448358e-05 0.01456260 0.01456260 -0.02850768 0.02857665
50 3.448358e-05 0.01458731 0.01458731 -0.02855613 0.02862509
51 3.448358e-05 0.01461123 0.01461123 -0.02860300 0.02867196
52 3.448358e-05 0.01463437 0.01463437 -0.02864835 0.02871732
53 3.448358e-05 0.01465676 0.01465676 -0.02869223 0.02876120
54 3.448358e-05 0.01467842 0.01467842 -0.02873470 0.02880366
55 3.448358e-05 0.01469939 0.01469939 -0.02877579 0.02884476
56 3.448358e-05 0.01471968 0.01471968 -0.02881556 0.02888453
57 3.448358e-05 0.01473932 0.01473932 -0.02885405 0.02892302
58 3.448358e-05 0.01475833 0.01475833 -0.02889130 0.02896027
59 3.448358e-05 0.01477672 0.01477672 -0.02892736 0.02899633
60 3.448358e-05 0.01479453 0.01479453 -0.02896227 0.02903123
61 3.448358e-05 0.01481177 0.01481177 -0.02899605 0.02906502
62 3.448358e-05 0.01482846 0.01482846 -0.02902876 0.02909772
63 3.448358e-05 0.01484461 0.01484461 -0.02906042 0.02912938
64 3.448358e-05 0.01486025 0.01486025 -0.02909107 0.02916003
65 3.448358e-05 0.01487539 0.01487539 -0.02912074 0.02918971
66 3.448358e-05 0.01489005 0.01489005 -0.02914947 0.02921844
67 3.448358e-05 0.01490424 0.01490424 -0.02917728 0.02924625
68 3.448358e-05 0.01491798 0.01491798 -0.02920422 0.02927318
69 3.448358e-05 0.01493128 0.01493128 -0.02923029 0.02929926
70 3.448358e-05 0.01494417 0.01494417 -0.02925554 0.02932451
71 3.448358e-05 0.01495664 0.01495664 -0.02927999 0.02934896
72 3.448358e-05 0.01496872 0.01496872 -0.02930367 0.02937263
73 3.448358e-05 0.01498042 0.01498042 -0.02932659 0.02939556
74 3.448358e-05 0.01499174 0.01499174 -0.02934880 0.02941776
75 3.448358e-05 0.01500271 0.01500271 -0.02937030 0.02943926
76 3.448358e-05 0.01501334 0.01501334 -0.02939112 0.02946009
77 3.448358e-05 0.01502363 0.01502363 -0.02941128 0.02948025
78 3.448358e-05 0.01503359 0.01503359 -0.02943081 0.02949978
79 3.448358e-05 0.01504324 0.01504324 -0.02944973 0.02951870
80 3.448358e-05 0.01505259 0.01505259 -0.02946805 0.02953702
81 3.448358e-05 0.01506164 0.01506164 -0.02948579 0.02955476
82 3.448358e-05 0.01507041 0.01507041 -0.02950298 0.02957194
83 3.448358e-05 0.01507890 0.01507890 -0.02951962 0.02958859
84 3.448358e-05 0.01508713 0.01508713 -0.02953575 0.02960471
85 3.448358e-05 0.01509510 0.01509510 -0.02955136 0.02962033
86 3.448358e-05 0.01510281 0.01510281 -0.02956649 0.02963546
87 3.448358e-05 0.01511029 0.01511029 -0.02958114 0.02965011
88 3.448358e-05 0.01511753 0.01511753 -0.02959534 0.02966430
89 3.448358e-05 0.01512455 0.01512455 -0.02960908 0.02967805
90 3.448358e-05 0.01513134 0.01513134 -0.02962240 0.02969137
91 3.448358e-05 0.01513793 0.01513793 -0.02963530 0.02970427
92 3.448358e-05 0.01514430 0.01514430 -0.02964780 0.02971677
93 3.448358e-05 0.01515048 0.01515048 -0.02965991 0.02972888
94 3.448358e-05 0.01515646 0.01515646 -0.02967164 0.02974061
95 3.448358e-05 0.01516226 0.01516226 -0.02968300 0.02975197
96 3.448358e-05 0.01516788 0.01516788 -0.02969401 0.02976298
97 3.448358e-05 0.01517332 0.01517332 -0.02970467 0.02977364
98 3.448358e-05 0.01517859 0.01517859 -0.02971500 0.02978397
99 3.448358e-05 0.01518370 0.01518370 -0.02972501 0.02979398
100 3.448358e-05 0.01518864 0.01518864 -0.02973471 0.02980368