Basic Time Series Analysis
Sudiksha Joshi
3/01/2019
Autocorrelation Function (ACF) and Partial ACF
Autocorrelation measures the linear relationship between lagged variables in a time series data. The ACF plot shows different autocorrelation coefficients. For example, \(r_1\) measures the relationship between y_t and y_(t-1). \(r_2\) measures the relationship between y_t and y_(t-2) and so on.
ACF and PACF plots measure the relationship between y_t and y_(t-k) after removing the effects of lags 1,2,…,(k-1). So the first partial autocorrelation is identical to the first autocorrelation, because there is nothing between them to remove. Each partial autocorrelation can be estimated as the last coefficient in an autoregressive model.
When a plot has trends, then the ACF decreases gradually as lags increase. Because the housing starts series has autocorrelation, it is not white noise. A time series is white noise if all the variables are independently and identically distributed with a mean of 0, and a constant variance. The blue lines in the plot indicate significane. The spikes in the plot that exceed the significane lines above and below imply that the current level of housing starts is significantly autocorrelated with its lagged values.
PACF
The partial auto-correlation function measures the correlation between current variable and lagged variable after eliminating the correlation from previous lags. In simple terms, the PACF removes the lags that cause autocorrelation.
From the plot below, we will include 3 lags. Adding more lags decreases the degrees of freedom and power as we add more regressors to the model.
Portmanteau tests for autocorrelation
ARIMA models explain or capture serial correlation present within a time series.
We test whether the first h autocorrelations are significantly different from what would be expected from a white noise process. A test for a group of autocorrelations is called a portmanteau test. We can do the Ljung-Box test.
H0: at each lag, the time series data points are i.i.d i.e. there is no autocorrelation.
Ha: data points at each lag are not i.i.d. and are serially correlated.
Stationarity
A time series is stationary whose properties are independent of the time in which we observe the data. So, a series with trends or seasonality is non-stationary as trends and seasonality change the values of parameters at different points in time. Alternatively, a white noise series is stationary as it looks the same any time.Generally, stationary time series have no predictable patterns in the long run. Such time plots will be approximately horizontal,have constant variance and mean, albeit they may be cyclical.
Strictly Stationary Series
When the distribution of elements x_(t_1),…,x_(t_n) is equal to that of x_(t_(1+m)),…,x_(t_(n+m)), ∀ t_i,m, then the time series model, {x_t}, is strictly stationary. The distribution of the time series should be constant even when time arbitrarily changes.
Differencing to make a stationary time series
We can make a non-stationary time series stationary by differencing consecutive observations. Lograrithmic transformations and differencing can stabalize the variance and mean of the time series,respectively.Furthermore, differencing eliminates seasonality and trends.
We can look at ACF plot to identify non-stationary time series. For suchlike data, ACF plummets to zero fast. On the contrary, the ACF of a unit root series decreases relatively gradually. The value of r_1 is often large and positive for non-stationary data.
##
## Box-Ljung test
##
## data: diff(time_ser[, 1])
## X-squared = 70.293, df = 10, p-value = 3.892e-11The null hypothesis is that the series is i.i.d. or has no serial correlation
The ACF of the differenced housing starts looks doesn’t like that of a white noise series. There are autocorrelations lying outside the 95% limits, and the Ljung-Box Q∗ statistic has a very small p-value of 3.892e-11 (for h=10). This suggests that the daily change in the US housing starts is not a random amount which is correlated with that of previous months.
Random Walk Model
Used for non-stationary economic and financial data, random walk models have long periods of trends and can change unpredictably in any direction. Thus, the forecasts are equal to the last observation as the values are equally likely to move up or down.
Let a time series be {w_t:t=1,…n}. If the elements of the series, w_i, are independent and identically distributed (i.i.d.), with zero mean, variance σ^2 and no serial correlation (i.e. Cor(w_i,w_j)≠ 0,∀ i≠j) then the time series is discrete white noise (DWN).
In particular, if the values w_i are drawn from a standard normal distribution (i.e. w_t ∼ N(0,σ^2)), then the series is known as Gaussian White Noise.
In a random walk, each term, x_t depends entirely on the previous term, x_(t−1) and a stochastic white noise term, w_t:
x_t = x_(t−1) + w_t
An extention of the random walk is the autoregressive model as it incorporates terms further back in time. Thus the AR model is linearly dependent on the previous terms.
Autoregressive Model of order p
A time series model, {x_t}, is an autoregressive model of order p, AR(p), if:
x_t = α_1 * x_(t−1) +… + α_p * x_(t−p) + w_t, where {w_t} is white noise.
Moving Average Model of order q
MA is a linear combination of the past white noise terms.
Intuitively, this means that the MA model sees such random white noise “shocks” directly at each current value of the model. This is in contrast to an AR(p) model, where the white noise “shocks” are only seen indirectly, via regression onto previous terms of the series.
A time series model, {x_t}, is a moving average model of order q, MA(q), if:
x_t = w_t + β_1 * w_(t−1) +…+ β_q * w_(t−q), where {w_t} is white noise
Autoregressive Moving Average Model of order p, q
A time series model, {x_t}, is an autoregressive moving average model of order p,q, ARMA(p,q), if:
x_t = α_1 * x_(t−1) + α_2 * x_(t−2) +…+ w_t + β_1 * w_(t−1) + β_2* w_(t−2) …+ β_q * w_(t−q)
Where {w_t} is white noise with E(w_t) = 0 and variance σ^2.
The former AR model considers its own past behaviour as inputs for the model and as such attempts to capture market participant effects, such as momentum and mean-reversion in stock trading.
The latter model is used to characterise “shock” information to a series, such as a surprise earnings announcement or unexpected event (such as the BP Deepwater Horizon oil spill).
Applying a difference operator to a non-stationary or a random walk series {x_t} gives a stationary or a white noise {w_t} series.
∇x_t=x_t − x_(t−1) = w_t
ARIMA repeatedly differences d times to make a stationary series.
Unit root or non-stationarity tests
We can use the statistical hypothesis unit toot tests to objectively determine whether the series requires differencing. In our analysis, we use the Augmented Dickey Fuller test.
The time series is modeled as: z_t = α * z_(t−1) + w_t, wherein w_t is discrete white noise. The null hypothesis is that α = 1, while the alternative hypothesis is that α < 1. In this test, the null hypothesis is that the data is not stationary or it is unit root, and we look for evidence that the null hypothesis is false. Consequently, small p-values (e.g., > 0.05) suggest that differencing is required.
##
## Augmented Dickey-Fuller Test
##
## data: time_ser[, 1]
## Dickey-Fuller = -2.4706, Lag order = 7, p-value = 0.3791
## alternative hypothesis: stationaryThe test statistic is much bigger than the 5% critical value, so we fail to reject the null hypothesis. That is, the data is not stationary. We can difference the data twice (2nd difference), and apply the test again.Occasionally the differenced data will not appear to be stationary and it may be necessary to difference the data a second time to obtain a stationary series.
##
## Augmented Dickey-Fuller Test
##
## data: diff(time_ser[, 1])
## Dickey-Fuller = -7.7858, Lag order = 7, p-value = 0.01
## alternative hypothesis: stationaryThis time, the test statistic is tiny, and well within the range we would expect for stationary data. So we can conclude that the differenced data are stationary.
Non Seasonal ARIMA Models
Combining autoregresion with differencing and a mocing average model yields a non-seasonal Auto-regressive Integrated Moving Average (ARIMA) model. A series with ARIMA(0,0,0) is a white noise series.
Intuitively, ARIMA denotes the number of previous time steps the current value of our variable depends on. For example, at time T, our variable X_t depends on X_(t-1) and X_(t-2), linearly. In this case, we have 2 AR terms and hence our p parameter=2
MA – This term is a measure of the average over multiple time periods we take into account. For example, to calculate the value of our variable at the current time step, if we take an average over previous 2 timesteps, the number of MA terms, denoted by q=2
Checking for stationarity of the predictors through plots
Graphically, all the predictors are non-stationary.Another way of checking is the unit root test for stationarity using the Augmented Dickey Fuller test for stationarity. The null hypothesis is that the series is unit root or non-stationarity.
The order of intergation is another concept closely associated to stationarity. The order tells the number of time we should difference the series to make it stationary.An I(0) series has order 0 if it does not require any differencing, and is already stationary. A series of order 1 or I(1) if it is non-stationary in the beginning, and the first difference makes it stationary. An I(0) series frequently crosses the mean, whereas I(1) and I(2) series can stary or wander farther from their mean value and rarely comes across the mean.
| statistic | p.value | parameter | method | alternative | |
|---|---|---|---|---|---|
| Housing_Starts | -2.4706305 | 0.3790981 | 7 | Augmented Dickey-Fuller Test | stationary |
| Income | -1.8670599 | 0.6345780 | 7 | Augmented Dickey-Fuller Test | stationary |
| Federal_funds_rate | -3.0956900 | 0.1145223 | 7 | Augmented Dickey-Fuller Test | stationary |
| Yield_spread | -2.7597494 | 0.2567196 | 7 | Augmented Dickey-Fuller Test | stationary |
| Securitized_consumer_loans | -0.6996825 | 0.9705642 | 7 | Augmented Dickey-Fuller Test | stationary |
| Unemployment_rate | -3.2116336 | 0.0859251 | 7 | Augmented Dickey-Fuller Test | stationary |
| CPI | -2.7648175 | 0.2545744 | 7 | Augmented Dickey-Fuller Test | stationary |
| Private_house_completed | -1.6420346 | 0.7298269 | 7 | Augmented Dickey-Fuller Test | stationary |
| Mortgage_rate | -3.3027467 | 0.0702159 | 7 | Augmented Dickey-Fuller Test | stationary |
| Real_estate_loans | -1.9457001 | 0.6012911 | 7 | Augmented Dickey-Fuller Test | stationary |
| House_supply | -2.5655219 | 0.3389323 | 7 | Augmented Dickey-Fuller Test | stationary |
All the series are stationary after differencing upto 3 times.
| statistic | p.value | parameter | method | alternative | |
|---|---|---|---|---|---|
| Housing_Starts | -7.785817 | 0.01 | 7 | Augmented Dickey-Fuller Test | stationary |
| Income | -15.757528 | 0.01 | 7 | Augmented Dickey-Fuller Test | stationary |
| Federal_funds_rate | -7.247981 | 0.01 | 7 | Augmented Dickey-Fuller Test | stationary |
| Yield_spread | -7.480514 | 0.01 | 7 | Augmented Dickey-Fuller Test | stationary |
| Securitized_consumer_loans | -12.691731 | 0.01 | 7 | Augmented Dickey-Fuller Test | stationary |
| Unemployment_rate | -4.733799 | 0.01 | 7 | Augmented Dickey-Fuller Test | stationary |
| CPI | -15.485510 | 0.01 | 7 | Augmented Dickey-Fuller Test | stationary |
| Private_house_completed | -6.393007 | 0.01 | 7 | Augmented Dickey-Fuller Test | stationary |
| Mortgage_rate | -7.348586 | 0.01 | 7 | Augmented Dickey-Fuller Test | stationary |
| Real_estate_loans | -13.229879 | 0.01 | 7 | Augmented Dickey-Fuller Test | stationary |
| House_supply | -8.092979 | 0.01 | 7 | Augmented Dickey-Fuller Test | stationary |
New dataframe with the differenced variables
Autoregressive Integrated Moving Average with Explanatory Variable (ARIMAX) Model
The standard ARIMA models forecast solely based on the past values of the housing starts, and does not have covariates. The model assumes that the future values are linearly dependent on the past values and previous stochastic shocks. Similar to ARIMA and a multivariate regression model is the ARIMAX model, wherein covariates are present on the right hand side of the model. Below is an ARIMAX model where x_t is a covariate at time t and a is its coefficient:
x_t = ax_t + α_1 x_(t−1) + α_2 * x_(t−2) +…+ w_t + β_1 * w_(t−1) + β_2* w_(t−2) …+ β_q * w_(t−q)
Where {w_t} is white noise with E(w_t) = 0 and variance σ^2
## Series: time_ser_diff[, 1]
## Regression with ARIMA(0,1,1) errors
##
## Coefficients:
## ma1 mortgR income_d2 sec_conL_d2 CPI_d3
## -0.4143 -56.5011 0.017 0.1658 -6.9073
## s.e. 0.0391 12.4954 0.031 0.2555 4.5430
## pvt_house_comp_d1 real_estL_d2
## 0.0931 -0.0927
## s.e. 0.0375 0.2098
##
## sigma^2 estimated as 10276: log likelihood=-3059.22
## AIC=6134.43 AICc=6134.72 BIC=6168.31
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -2.505748 100.8716 75.43931 -0.6482344 5.694265 0.3893491
## ACF1
## Training set -0.006246253y_t = c + -0.4143ε_(t-1) + -56.5011mortgR + 0.017* income_d2 + 0.1658 * sec_conL_d2 + -6.9073* CPI_d3 + 0.0931 * pvt_house_comp_d1 + -0.0927 * real_estL_d2
ARIMA(0,1,1) is also an MA(1) model, where the coefficient of ε_(t-1) tells how quickly the forecasts converge to the mean. From th plot of forecasts, when the blue line is horizontal, it means that the forecasts have converged to the mean.
the ARIMA errors that should resemble a white noise series.
##
## Ljung-Box test
##
## data: Residuals from Regression with ARIMA(0,1,1) errors
## Q* = 26.644, df = 17, p-value = 0.06351
##
## Model df: 7. Total lags used: 24H_o = no autocorrelation in the residuals.
The results Ljung-Box test are insignificant (i.e., the p-values = 0.0635 is big). Thus, we can conclude that the residuals are not serially correlated, producing precise coverage of the prediction intervals.
The time plot and histogram of the residuals shows that the variance in the residuals are almost constant.
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Jan 2019 NA NA NA NA NA
## Feb 2019 NA NA NA NA NA
## Mar 2019 NA NA NA NA NA
## Apr 2019 962.9494 777.897172 1148.002 679.93643 1245.962
## May 2019 967.4201 767.336823 1167.503 661.41915 1273.421
## Jun 2019 977.9970 763.935586 1192.058 650.61832 1305.376
## Jul 2019 975.0191 747.838005 1202.200 627.57559 1322.463
## Aug 2019 977.6590 738.075505 1217.242 611.24768 1344.070
## Sep 2019 996.9350 745.560391 1248.310 612.49070 1381.379
## Oct 2019 982.2730 719.636036 1244.910 580.60442 1383.942
## Nov 2019 965.7380 692.302183 1239.174 547.55399 1383.922
## Dec 2019 974.8773 691.053201 1258.701 540.80579 1408.949
## Jan 2020 975.7839 681.938482 1269.629 526.38614 1425.182
## Feb 2020 970.9937 667.457773 1274.530 506.77554 1435.212
## Mar 2020 963.9266 650.999969 1276.853 485.34663 1442.507
## Apr 2020 986.4283 664.384774 1308.472 493.90523 1478.951
## May 2020 947.8049 616.895540 1278.714 441.72271 1453.887
## Jun 2020 975.3698 635.825987 1314.914 456.08237 1494.657
## Jul 2020 952.5094 604.545400 1300.473 420.34438 1484.674
## Aug 2020 972.2254 616.040121 1328.411 427.48704 1516.964
## Sep 2020 949.9511 585.730156 1314.172 392.92323 1506.979
## Oct 2020 955.9691 583.885922 1328.052 386.91699 1525.021
## Nov 2020 946.6201 566.837510 1326.403 365.79272 1527.448
## Dec 2020 929.7877 542.458617 1317.117 337.41898 1522.156
## Jan 2021 928.6303 533.898990 1323.362 324.94086 1532.320
## Feb 2021 924.4700 522.472736 1326.467 309.66825 1539.272
## Mar 2021 919.8406 510.706437 1328.975 294.12390 1545.557
## Apr 2021 913.6917 497.542988 1329.840 277.24718 1550.136
## May 2021 909.3733 486.326375 1332.420 262.37886 1556.368
## Jun 2021 900.0800 470.245560 1329.915 242.70493 1557.455
## Jul 2021 881.5941 445.077625 1318.111 213.99975 1549.188
## Aug 2021 882.8924 439.794678 1325.990 205.23289 1560.552
## Sep 2021 882.5096 432.926866 1332.092 194.93216 1570.087
## Oct 2021 891.2318 435.256376 1347.207 193.87757 1588.586
## Nov 2021 868.9268 406.647029 1331.206 161.93092 1575.923
## Dec 2021 879.3924 410.893175 1347.892 162.88467 1595.900
## Jan 2022 833.0272 358.389941 1307.664 107.13217 1558.922
## Feb 2022 836.0964 355.399568 1316.793 100.93402 1571.259
## Mar 2022 846.1628 359.481780 1332.844 101.84839 1590.477
## Apr 2022 850.1969 357.604372 1342.789 96.84163 1603.552
## May 2022 798.1961 299.762156 1296.630 35.90716 1560.485
## Jun 2022 752.1597 247.951995 1256.367 -18.95942 1523.279
## Jul 2022 738.0988 228.182810 1248.015 -41.75042 1517.948
## Aug 2022 736.0745 220.513285 1251.636 -52.40831 1524.557
## Sep 2022 735.7561 214.610878 1256.901 -61.26671 1532.779
## Oct 2022 595.9416 69.271604 1122.612 -209.53064 1401.414
## Nov 2022 555.4955 23.358071 1087.633 -258.33847 1369.329
## Dec 2022 642.1332 104.583957 1179.683 -179.97744 1464.244
## Jan 2023 748.1898 205.282584 1291.097 -82.11511 1578.495
## Feb 2023 786.7645 238.551825 1334.977 -51.65445 1625.184
## Mar 2023 749.0456 195.578261 1302.513 -97.40967 1595.501
## Apr 2023 711.7161 153.043453 1270.389 -142.69998 1566.132
## May 2023 692.2414 128.411572 1256.071 -170.06192 1554.545
## Jun 2023 670.8026 101.862290 1239.743 -199.31651 1540.922
## Jul 2023 642.9208 68.915490 1216.926 -234.94454 1520.786
## Aug 2023 617.1178 38.091920 1196.144 -268.42589 1502.662
## Sep 2023 627.5304 43.526965 1211.534 -265.62577 1520.687
## Oct 2023 600.7415 11.802669 1189.680 -299.96272 1501.446
## Nov 2023 597.6188 3.785510 1191.452 -310.57083 1505.808
## Dec 2023 522.1386 -76.549081 1120.826 -393.47519 1437.752
## Jan 2024 536.2431 -67.259901 1139.746 -386.73510 1459.221
## Feb 2024 519.4670 -88.813297 1127.747 -410.81741 1449.751
## Mar 2024 485.0019 -128.018396 1098.022 -452.53172 1422.535
## Apr 2024 441.2766 -176.447283 1059.001 -503.45056 1386.004
## May 2024 431.9446 -190.447412 1054.337 -519.92183 1383.811
## Jun 2024 446.0354 -180.989957 1073.061 -512.91712 1404.988
## Jul 2024 520.5763 -111.048332 1152.201 -445.41024 1486.563
## Aug 2024 480.5456 -155.645224 1116.736 -492.42428 1453.515
## Sep 2024 464.9076 -175.816768 1105.632 -514.99575 1444.811
## Oct 2024 505.4676 -139.758427 1150.694 -481.32047 1492.256
## Nov 2024 514.5982 -135.098400 1164.295 -479.02699 1508.223
## Dec 2024 532.3605 -121.776067 1186.497 -468.05503 1532.776
## Jan 2025 517.5168 -141.029727 1176.063 -489.64322 1524.677
## Feb 2025 534.0503 -128.876993 1196.978 -479.80949 1547.910
## Mar 2025 546.5375 -120.741664 1213.817 -473.97794 1567.053
## Apr 2025 594.5617 -77.041241 1266.165 -432.56636 1621.690
## May 2025 651.1825 -24.716490 1327.082 -382.51582 1684.881
## Jun 2025 697.8912 17.723243 1378.059 -342.33593 1738.118
## Jul 2025 696.2736 11.863339 1380.684 -350.44158 1742.989
## Aug 2025 729.6469 41.020415 1418.273 -323.51642 1782.810
## Sep 2025 732.7260 39.909000 1425.543 -326.84616 1792.298
## Oct 2025 750.5672 53.584862 1447.550 -315.37529 1816.510
## Nov 2025 746.6843 45.561400 1447.807 -325.59065 1818.959
## Dec 2025 772.5303 67.291110 1477.769 -306.03996 1851.101
## Jan 2026 747.2194 37.887796 1456.551 -337.60966 1832.048
## Feb 2026 716.6495 3.249029 1430.050 -374.40238 1807.701
## Mar 2026 719.1172 1.670827 1436.563 -378.12232 1816.357
## Apr 2026 674.4882 -46.981269 1395.958 -428.90414 1777.881
## May 2026 714.0429 -11.427383 1439.513 -395.46817 1823.554
## Jun 2026 701.0543 -28.394955 1430.503 -414.54204 1816.651
## Jul 2026 716.2362 -17.170300 1449.643 -405.41226 1837.885
## Aug 2026 719.5982 -17.744423 1456.941 -408.07002 1847.266
## Sep 2026 725.0384 -16.219390 1466.296 -408.61755 1858.694
## Oct 2026 716.7030 -28.449357 1461.855 -422.90919 1856.315
## Nov 2026 705.2021 -43.824623 1454.229 -440.33542 1850.740
## Dec 2026 689.9120 -62.969142 1442.793 -461.52034 1841.344
## Jan 2027 652.9294 -103.786488 1409.645 -504.36770 1810.227
## Feb 2027 637.4100 -123.121400 1397.941 -525.72238 1800.542
## Mar 2027 655.0844 -109.243378 1419.412 -513.85405 1824.023
## Apr 2027 654.6378 -113.467620 1422.743 -520.07806 1829.354
## May 2027 670.0284 -101.836140 1441.893 -510.43655 1850.493
## Jun 2027 700.4244 -75.181087 1476.030 -485.76182 1886.611
## Jul 2027 726.7149 -52.613518 1506.043 -465.16508 1918.595
## Aug 2027 732.5215 -50.512266 1515.555 -465.02528 1930.068
## Sep 2027 748.8399 -37.881621 1535.561 -454.34685 1952.027
## Oct 2027 725.1997 -65.192459 1515.592 -483.60079 1934.000
## Nov 2027 720.0649 -73.980900 1514.111 -494.32335 1934.453
## Dec 2027 742.9036 -54.779071 1540.586 -477.04679 1962.854
## Jan 2028 786.5131 -14.789997 1587.816 -438.97424 2012.000
## Feb 2028 785.6068 -19.300386 1590.514 -445.39254 2016.606
## Mar 2028 779.9123 -28.582965 1588.408 -456.57451 2016.399
## Apr 2028 794.0816 -17.985889 1606.149 -447.86845 2036.032
## May 2028 760.4421 -55.181895 1576.066 -486.94718 2007.831
## Jun 2028 822.5043 3.339186 1641.669 -430.30065 2075.309
## Jul 2028 833.7743 11.083253 1656.465 -424.42307 2091.972
## Aug 2028 855.3708 29.168922 1681.573 -408.19592 2118.937
## Sep 2028 871.8577 42.159867 1701.556 -397.05563 2140.771
## Oct 2028 900.0491 66.870024 1733.228 -374.18836 2174.287
## Nov 2028 898.1512 61.505239 1734.797 -381.38837 2177.691
## Dec 2028 907.6833 67.584839 1747.782 -377.13641 2192.503
## Jan 2029 855.8142 12.277347 1699.351 -434.26407 2145.892
## Feb 2029 884.7989 37.837690 1731.760 -410.51651 2180.114
## Mar 2029 895.3687 44.996849 1745.741 -405.16283 2195.900
## Apr 2029 900.5542 46.785348 1754.323 -405.17259 2206.281
## May 2029 915.4504 58.298000 1772.603 -395.45108 2226.352
## Jun 2029 920.9223 60.399645 1781.445 -395.13353 2236.978
## Jul 2029 947.2545 83.374780 1811.134 -373.93553 2268.445
## Aug 2029 952.5624 85.338527 1819.786 -373.74204 2278.867
## Sep 2029 952.7161 82.161021 1823.271 -378.68301 2284.115
## Oct 2029 950.4926 76.618954 1824.366 -385.98181 2286.967
## Nov 2029 915.2846 38.104962 1792.464 -426.24589 2256.815
## Dec 2029 879.2989 -1.174300 1759.772 -467.26867 2225.867
## Jan 2030 860.9671 -22.787378 1744.722 -490.61877 2212.553
## Feb 2030 897.8160 10.792320 1784.840 -458.76966 2254.402
## Mar 2030 878.6726 -11.608242 1768.953 -482.89446 2240.240
## Apr 2030 853.3570 -40.169200 1746.883 -513.17337 2219.887
## May 2030 834.2240 -62.535680 1730.984 -537.25159 2205.700
## Jun 2030 867.0968 -32.884789 1767.078 -509.30629 2243.500
## Jul 2030 876.0634 -27.128630 1779.256 -505.24963 2257.377
## Aug 2030 871.4223 -34.968789 1777.813 -514.78327 2257.628
## Sep 2030 905.1034 -4.475533 1814.682 -485.97754 2296.184
## Oct 2030 921.1424 8.386724 1833.898 -474.79692 2317.082
## Nov 2030 893.4805 -22.440831 1809.402 -507.30027 2294.261
## Dec 2030 872.4361 -46.639958 1791.512 -533.16943 2278.042
## Jan 2031 878.2750 -43.944972 1800.495 -532.13876 2288.689
## Feb 2031 884.0083 -41.345043 1809.362 -531.19749 2299.214
## Mar 2031 874.1335 -54.342500 1802.610 -545.84801 2294.115
## Apr 2031 875.8631 -55.725208 1807.451 -548.87824 2300.604
## May 2031 889.7637 -44.926402 1824.454 -539.72147 2319.249
## Jun 2031 877.9175 -59.864257 1815.699 -556.29594 2312.131
## Jul 2031 884.3733 -56.489936 1825.237 -554.55284 2323.299
## Aug 2031 866.6443 -77.290313 1810.579 -576.97912 2310.268
## Sep 2031 871.8708 -75.125292 1818.867 -576.43473 2320.176
## Oct 2031 831.6862 -118.361420 1781.734 -621.28627 2284.659
## Nov 2031 852.9890 -100.100430 1806.078 -604.63551 2310.614
## Dec 2031 857.8825 -98.239073 1814.004 -604.37927 2320.144
## Jan 2032 884.5670 -74.577093 1843.711 -582.31732 2351.451
## Feb 2032 914.5188 -47.638335 1876.676 -556.97358 2386.011
## Mar 2032 912.5993 -52.561513 1877.760 -563.48679 2388.685
## Apr 2032 885.1225 -83.032670 1853.278 -595.54304 2365.788
## May 2032 908.5233 -62.616977 1879.663 -576.70756 2393.754
## Jun 2032 934.1199 -39.996245 1908.236 -555.66220 2423.902
## Jul 2032 904.1784 -72.904631 1881.261 -590.14116 2398.498
## Aug 2032 924.5788 -55.462162 1904.620 -574.26451 2423.422
## Sep 2032 889.0490 -93.940919 1872.039 -614.30437 2392.402
## Oct 2032 883.9165 -102.013639 1869.847 -623.93353 2391.767
## Nov 2032 886.9133 -101.948238 1875.775 -625.41993 2399.247
## Dec 2032 877.9258 -113.858548 1869.710 -638.87746 2394.729
## Jan 2033 884.2713 -110.427206 1878.970 -636.98880 2405.531
## Feb 2033 912.1055 -85.498661 1909.710 -613.59842 2437.809
## Mar 2033 895.0490 -105.452458 1895.550 -635.08592 2425.184
## Apr 2033 901.4257 -101.964551 1904.816 -633.12729 2435.979
## May 2033 890.2427 -116.028175 1896.514 -648.71579 2429.201
## Jun 2033 906.8079 -102.335350 1915.951 -636.54350 2450.159
## Jul 2033 911.0623 -100.945143 1923.070 -636.66951 2458.794
## Aug 2033 924.9145 -89.949043 1939.778 -627.18534 2477.014
## Sep 2033 939.2285 -78.483161 1956.940 -617.22715 2495.684
## Oct 2033 939.8879 -80.663871 1960.440 -620.91135 2500.687
## Nov 2033 924.0575 -99.326530 1947.442 -641.07332 2489.188
## Dec 2033 933.8500 -92.358544 1960.058 -635.60051 2503.300
## Jan 2034 931.3677 -97.657470 1960.393 -642.39050 2505.126
## Feb 2034 927.9981 -103.836128 1959.832 -650.05616 2506.052
## Mar 2034 943.7561 -90.879510 1978.392 -638.58251 2526.095
## Apr 2034 978.4052 -59.024166 2015.835 -608.20612 2565.017
## May 2034 957.9093 -82.306459 1998.125 -632.96340 2548.782
## Jun 2034 968.7692 -74.225396 2011.764 -626.35338 2563.892
## Jul 2034 994.3672 -51.398924 2040.133 -604.99405 2593.728
## Aug 2034 1000.5827 -47.947517 2049.113 -603.00590 2604.171
## Sep 2034 974.9844 -76.302764 2026.272 -632.82056 2582.789
## Oct 2034 967.7883 -86.248472 2021.825 -644.22186 2579.799
## Nov 2034 968.5929 -88.186485 2025.372 -647.61168 2584.797
## Dec 2034 990.0772 -69.437549 2049.592 -630.31079 2610.465
## Jan 2035 989.3911 -72.852090 2051.634 -635.16966 2613.952
## Feb 2035 1017.1325 -47.832100 2082.097 -611.59029 2645.855
## Mar 2035 1010.5042 -57.174854 2078.183 -622.36999 2643.378
## Apr 2035 1019.4764 -50.910233 2089.863 -617.53867 2656.491
## May 2035 1012.5017 -60.585648 2085.589 -628.64377 2653.647
## Jun 2035 1014.1540 -61.627290 2089.935 -631.11151 2659.420
## Jul 2035 1008.0298 -70.438750 2086.498 -641.34551 2657.405
## Aug 2035 1001.1580 -79.991056 2082.307 -652.31682 2654.633
## Sep 2035 1052.8470 -30.975983 2136.670 -604.71723 2710.411
## Oct 2035 1031.9094 -54.580948 2118.400 -629.73421 2693.553
## Nov 2035 1058.1873 -30.963920 2147.338 -607.52573 2723.900
## Dec 2035 1039.4285 -52.376959 2131.234 -630.34388 2709.201
## Jan 2036 1061.0907 -33.362680 2155.544 -612.73131 2734.913
## Feb 2036 1051.8507 -45.244238 2148.946 -626.01120 2729.713
## Mar 2036 1084.6916 -15.038406 2184.422 -597.20033 2766.584
## Apr 2036 1072.1032 -30.255723 2174.462 -613.80928 2758.016
## May 2036 1087.8809 -17.100622 2192.862 -602.04250 2777.804
## Jun 2036 1070.3832 -37.214717 2177.981 -623.54163 2764.308
## Jul 2036 1073.2131 -36.994987 2183.421 -624.70367 2771.130
## Aug 2036 1061.1373 -51.674922 2173.950 -640.76213 2763.037
## Sep 2036 1076.4265 -38.983759 2191.837 -629.44628 2782.299
## Oct 2036 1030.8675 -87.134766 2148.870 -678.96940 2740.704
## Nov 2036 1012.6975 -107.890708 2133.286 -701.09429 2726.489
## Dec 2036 988.1812 -134.987095 2111.349 -729.55646 2705.919
## Jan 2037 986.1051 -139.637335 2111.847 -735.56936 2707.779
## Feb 2037 978.3742 -149.936477 2106.685 -747.22805 2703.976
## Mar 2037 997.2146 -133.658428 2128.088 -732.30647 2726.736
## Apr 2037 991.3922 -142.037477 2124.822 -742.03891 2724.823
## May 2037 962.5235 -173.457063 2098.504 -774.80885 2699.856
## Jun 2037 946.8512 -191.674553 2085.377 -794.37366 2688.076
## Jul 2037 956.2307 -184.834479 2097.296 -788.87791 2701.339
## Aug 2037 955.1197 -188.479301 2098.719 -793.86406 2704.104
## Sep 2037 959.3362 -186.791061 2105.463 -793.51419 2712.187
## Oct 2037 1005.7721 -142.877871 2154.422 -750.93642 2762.481
## Nov 2037 989.0582 -162.108912 2140.225 -771.49996 2749.616
## Dec 2037 1026.2125 -127.466207 2179.891 -738.18685 2790.612
## Jan 2038 1031.7891 -124.395865 2187.974 -736.44321 2800.021
## Feb 2038 1050.6641 -108.021600 2209.350 -721.39277 2822.721
## Mar 2038 1017.7891 -143.392029 2178.970 -758.08418 2793.662
## Apr 2038 1042.7122 -120.958908 2206.383 -736.96921 2822.394
## May 2038 1049.4244 -116.731464 2215.580 -734.05710 2832.906
## Jun 2038 1060.7352 -107.900162 2229.370 -726.53833 2848.009
## Jul 2038 1045.8489 -125.260565 2216.958 -745.20849 2836.906
## Aug 2038 1088.2330 -85.345454 2261.812 -706.60038 2883.066
## Sep 2038 1070.8992 -105.143047 2246.942 -727.70222 2869.501
## Oct 2038 1038.9984 -139.502547 2217.499 -763.36325 2841.360
## Nov 2038 1018.5319 -162.422496 2199.486 -787.58201 2824.646
## Dec 2038 1028.9355 -154.467334 2212.338 -780.92297 2838.794
## Jan 2039 998.7042 -187.142054 2184.550 -814.89114 2812.300
## Feb 2039 1005.6143 -182.670347 2193.899 -811.71022 2822.939
## Mar 2039 1019.7942 -170.923734 2210.512 -801.25175 2840.840
## Apr 2039 993.9851 -199.161263 2187.131 -830.77480 2818.745
## May 2039 1026.1075 -169.462321 2221.677 -802.35876 2854.574
## Jun 2039 1043.3409 -154.647496 2241.329 -788.82425 2875.506
## Jul 2039 1048.4913 -151.910789 2248.893 -787.36527 2884.348
## Aug 2039 1017.7653 -185.045636 2220.576 -821.77528 2857.306
## Sep 2039 1053.2109 -152.004102 2258.426 -790.00637 2896.428
## Oct 2039 1015.6613 -191.952890 2223.276 -831.22524 2862.548
## Nov 2039 1009.3200 -200.688700 2219.329 -841.22862 2859.869
## Dec 2039 1018.8598 -193.538712 2231.258 -835.34369 2873.063
## Jan 2040 1025.1498 -189.633735 2239.933 -832.70129 2883.001
## Feb 2040 1050.2017 -166.962237 2267.366 -811.28989 2911.693
## Mar 2040 1047.8457 -171.693968 2267.385 -817.27927 2912.971
## Apr 2040 1061.8015 -160.109265 2283.712 -806.94976 2930.553
## May 2040 1055.9746 -168.302688 2280.252 -816.39595 2928.345
## Jun 2040 1065.0742 -161.565074 2291.713 -810.90868 2941.057
## Jul 2040 1071.9036 -157.093055 2300.900 -807.68461 2951.492
## Aug 2040 1061.1913 -170.158344 2292.541 -821.99545 2944.378
## Sep 2040 1086.7885 -146.909531 2320.487 -799.98982 2973.567
## Oct 2040 1070.1821 -165.859922 2306.224 -820.18103 2960.545
## Nov 2040 1071.2565 -167.125019 2309.638 -822.68459 2965.198
## Dec 2040 1060.6300 -180.086598 2301.347 -836.88231 2958.142
## Jan 2041 1079.5713 -163.476013 2322.619 -821.50553 2980.648
## Feb 2041 1081.4861 -163.887658 2326.860 -823.14868 2986.121
## Mar 2041 1079.3736 -168.322130 2327.069 -828.81236 2987.560
## Apr 2041 1084.7809 -165.232520 2334.794 -826.94967 2996.512
## May 2041 1087.5569 -164.769983 2339.884 -827.71179 3002.826
## Jun 2041 1098.6450 -155.991080 2353.281 -820.15528 3017.445
## Jul 2041 1075.0838 -181.857199 2332.025 -847.24155 2997.409
## Aug 2041 1103.1990 -156.042616 2362.441 -822.64488 3029.043
## Sep 2041 1075.0485 -186.489698 2336.587 -854.30766 3004.405
## Oct 2041 1089.2411 -174.589402 2353.072 -843.62085 3022.103
## Nov 2041 1063.8188 -202.299824 2329.937 -872.54256 3000.180
## Dec 2041 1085.3105 -183.092194 2353.713 -854.54403 3025.165
## Jan 2042 1040.7127 -229.970002 2311.395 -902.62877 2984.054
## Feb 2042 1029.0521 -243.906401 2302.011 -917.76994 2975.874
## Mar 2042 1025.0689 -250.161396 2300.299 -925.22755 2975.365
## Apr 2042 1030.4452 -247.052881 2307.943 -923.31951 2984.210
## May 2042 1027.6756 -252.086248 2307.437 -929.55123 2984.902
## Jun 2042 1032.0844 -249.937157 2314.106 -928.59838 2992.767
## Jul 2042 1026.5658 -257.711559 2310.843 -937.56691 2990.698
## Aug 2042 1000.3639 -286.165257 2286.893 -967.21265 2967.940
## Sep 2042 1009.6491 -279.127968 2298.426 -961.36531 2980.663
## Oct 2042 1005.4885 -285.532521 2296.509 -968.95775 2979.935
## Nov 2042 1011.0432 -282.217829 2304.304 -966.82888 2988.915
## Dec 2042 986.0252 -309.472084 2281.522 -995.26690 2967.317
## Jan 2043 988.2229 -309.506712 2285.952 -996.48326 2972.929
## Feb 2043 1015.0028 -284.955266 2314.961 -973.11152 3003.117
## Mar 2043 1024.7107 -277.472105 2326.894 -966.80604 3016.227
## Apr 2043 1008.3641 -296.039621 2312.768 -986.54923 3003.277
## May 2043 1039.7101 -266.910689 2346.331 -958.59398 3038.014
## Jun 2043 1029.9513 -278.882884 2338.786 -971.73786 3031.641
## Jul 2043 1052.0853 -258.958494 2363.129 -952.98318 3057.154
## Aug 2043 1064.0901 -249.159678 2377.340 -944.35211 3072.532
## Sep 2043 1087.3139 -228.138083 2402.766 -924.49630 3099.124
## Oct 2043 1068.7107 -248.939810 2386.361 -946.46186 3083.883
## Nov 2043 1075.8919 -243.953501 2395.737 -942.63745 3094.421
## Dec 2043 1056.1023 -265.934296 2378.139 -965.77821 3077.983
## Jan 2044 1089.9179 -234.306316 2414.142 -935.30828 3115.144
## Feb 2044 1063.9470 -262.461274 2390.355 -964.61937 3092.513
## Mar 2044 1075.3031 -253.285528 2403.892 -956.59787 3107.204
## Apr 2044 1071.3727 -259.392801 2402.138 -963.85749 3106.603
## May 2044 1114.7334 -218.205359 2447.672 -923.82051 3153.287
## Jun 2044 1082.4071 -252.701414 2417.516 -959.46516 3124.279
## Jul 2044 1084.0349 -253.239860 2421.310 -961.15033 3129.220
## Aug 2044 1069.7662 -269.671195 2409.204 -978.72654 3118.259
## Sep 2044 1085.7344 -255.862244 2427.331 -966.06062 3137.529
## Oct 2044 1062.8464 -280.906016 2406.599 -992.24558 3117.938
## Nov 2044 1080.5893 -265.315389 2426.494 -977.79432 3138.973
## Dec 2044 1099.8415 -248.212078 2447.895 -961.82855 3161.512
## Jan 2045 1079.0864 -271.112705 2429.285 -985.86491 3144.038
## Feb 2045 1103.0904 -249.250743 2455.431 -965.13688 3171.318
## Mar 2045 1125.4527 -229.027057 2479.932 -946.04534 3196.951
## Apr 2045 1119.8475 -236.767602 2476.463 -954.91624 3194.611
## May 2045 1119.5746 -239.172360 2478.322 -958.44957 3197.599
## Jun 2045 1141.2437 -219.631879 2502.119 -940.03590 3222.523
## Jul 2045 1123.9712 -239.029642 2486.972 -960.55871 3208.501
## Aug 2045 1131.5298 -233.593001 2496.653 -956.24537 3219.305
## Sep 2045 1144.5416 -222.699916 2511.783 -946.47384 3235.557
## Oct 2045 1149.2905 -220.066341 2518.647 -944.96009 3243.541
## Nov 2045 1147.2703 -224.198722 2518.739 -950.21056 3244.751
## Dec 2045 1158.2405 -215.337420 2531.818 -942.46563 3258.947
## Jan 2046 1164.2691 -211.414435 2539.953 -939.65731 3268.196
## Feb 2046 1157.7668 -220.019121 2535.553 -949.37496 3264.909
## Mar 2046 1105.9801 -273.905124 2485.865 -1004.37222 3216.332
## Apr 2046 1138.9723 -243.008940 2520.954 -974.58561 3252.530
## May 2046 1145.5913 -238.482842 2529.665 -971.16741 3262.350
## Jun 2046 1122.7446 -263.419220 2508.908 -997.21001 3242.699
## Jul 2046 1139.2137 -249.036607 2527.464 -983.93195 3262.359
## Aug 2046 1144.7564 -245.577398 2535.090 -981.57564 3271.088
## Sep 2046 1156.4540 -235.960108 2548.868 -973.05960 3285.968
## Oct 2046 1167.6007 -226.890635 2562.092 -965.08974 3300.291
## Nov 2046 1152.0001 -244.565282 2548.566 -983.86235 3287.863
## Dec 2046 1110.1998 -288.436625 2508.836 -1028.83004 3249.230
## Jan 2047 1116.4210 -284.283460 2517.125 -1025.77159 3258.614
## Feb 2047 1135.6346 -267.134714 2538.404 -1009.71596 3280.985
## Mar 2047 1136.9367 -267.894563 2541.768 -1011.56731 3285.441
## Apr 2047 1131.5496 -275.340564 2538.440 -1020.10321 3283.202
## May 2047 1152.0869 -256.859094 2561.033 -1002.71005 3306.884
## Jun 2047 1140.4603 -270.538586 2551.459 -1017.47626 3298.397
## Jul 2047 1174.1970 -238.851759 2587.246 -986.87458 3335.269
## Aug 2047 1125.9112 -289.184502 2541.007 -1038.29089 3290.113
## Sep 2047 1154.9727 -262.166916 2572.112 -1012.35531 3322.301
## Oct 2047 1129.1148 -290.065856 2548.295 -1041.33470 3299.564
## Nov 2047 1154.7325 -266.486218 2575.951 -1018.83396 3328.299
## Dec 2047 1167.4196 -255.834330 2590.673 -1009.25942 3344.099
## Jan 2048 1136.2332 -289.052925 2561.519 -1043.55383 3316.020
## Feb 2048 1132.5532 -294.762331 2559.869 -1050.33752 3315.444
## Mar 2048 1156.3971 -272.944848 2585.739 -1029.59279 3342.387
## Apr 2048 1131.5746 -299.790970 2562.940 -1057.51015 3320.659
## May 2048 1160.0329 -273.353488 2593.419 -1032.14240 3352.208
## Jun 2048 1100.7061 -334.698191 2536.110 -1094.55532 3295.968
## Jul 2048 1102.4360 -334.983400 2539.855 -1095.90725 3300.779
## Aug 2048 1144.1449 -295.286744 2583.577 -1057.27582 3345.566
## Sep 2048 1120.9743 -320.466775 2562.415 -1083.51959 3325.468
## Oct 2048 1122.8968 -320.550945 2566.345 -1084.66602 3330.460
## Nov 2048 1082.9135 -362.538110 2528.365 -1127.71397 3293.541
## Dec 2048 1088.6061 -358.846647 2536.059 -1125.08182 3302.294
## Jan 2049 1101.9330 -347.518050 2551.384 -1114.81108 3318.677
## Feb 2049 1076.0071 -375.439464 2527.454 -1143.78889 3295.803
## Mar 2049 1105.1905 -348.248970 2558.630 -1117.65334 3328.034
## Apr 2049 1132.4087 -323.020905 2587.838 -1093.47878 3358.296
## May 2049 1075.8674 -381.549547 2533.284 -1153.05948 3304.794
## Jun 2049 1121.9652 -337.436403 2581.367 -1109.99697 3353.927
## Jul 2049 1123.1962 -338.187485 2584.580 -1111.79725 3358.190
## Aug 2049 1123.4422 -339.920745 2586.805 -1114.57829 3361.463
## Sep 2049 1091.6792 -373.660411 2557.019 -1149.36433 3332.723
## Oct 2049 1130.0418 -337.271703 2597.355 -1114.02058 3374.104
## Nov 2049 1109.9930 -359.291844 2579.278 -1137.08427 3357.070
## Dec 2049 1113.0416 -358.211915 2584.295 -1137.04650 3363.130
## Jan 2050 1092.2596 -380.960003 2565.479 -1160.83535 3345.355
## Feb 2050 1096.2484 -378.934652 2571.431 -1159.84937 3352.346
## Mar 2050 1099.3212 -377.822652 2576.465 -1159.77537 3358.418
## Apr 2050 1086.1717 -392.930396 2565.274 -1175.91973 3348.263
## May 2050 1119.4037 -361.653959 2600.461 -1145.67854 3384.486
## Jun 2050 1112.3490 -370.661744 2595.360 -1155.72021 3380.418
## Jul 2050 1135.8782 -349.083033 2620.839 -1135.17402 3406.930
## Aug 2050 1136.5940 -350.315113 2623.503 -1137.43727 3410.625
## Sep 2050 1133.4925 -355.362032 2622.347 -1143.51401 3410.499
## Oct 2050 1121.2998 -369.497562 2612.097 -1158.67801 3401.278
## Nov 2050 1124.0110 -368.726637 2616.749 -1158.93422 3406.956
## Dec 2050 1144.3740 -350.301409 2639.049 -1141.53480 3430.283
## Jan 2051 1096.9762 -399.634552 2593.587 -1191.89242 3385.845
## Feb 2051 1115.6890 -382.854499 2614.233 -1176.13552 3407.514
## Mar 2051 1106.4114 -394.062420 2606.885 -1188.36527 3401.188
## Apr 2051 1123.9093 -378.492266 2626.311 -1173.81564 3421.634
## May 2051 1119.2194 -385.107519 2623.546 -1181.45010 3419.889
## Jun 2051 1140.6609 -365.588956 2646.911 -1162.94945 3444.271
## Jul 2051 1134.6401 -373.530165 2642.810 -1171.90727 3441.187
## Aug 2051 1165.3176 -344.770646 2675.406 -1144.16307 3474.798
## Sep 2051 1189.7247 -322.279093 2701.728 -1122.68555 3502.135
## Oct 2051 1201.2904 -312.626487 2715.207 -1114.04570 3516.627
## Nov 2051 1192.2613 -323.566298 2708.089 -1125.99698 3510.520
## Dec 2051 1195.1165 -322.619467 2712.852 -1126.06034 3516.293
## Jan 2052 1149.2599 -370.381968 2668.902 -1174.83177 3473.352
## Feb 2052 1203.6489 -317.896515 2725.194 -1123.35399 3530.652
## Mar 2052 1158.7103 -364.736267 2682.157 -1171.20014 3488.621
## Apr 2052 1188.8810 -336.464299 2714.226 -1143.93333 3521.695
## May 2052 1184.9618 -342.279938 2712.204 -1150.75287 3520.676
## Jun 2052 1199.1483 -329.987458 2728.284 -1139.46305 3537.760
## Jul 2052 1199.2028 -331.824663 2730.230 -1142.30167 3540.707
## Aug 2052 1171.2991 -361.617813 2704.216 -1173.09500 3515.693
## Sep 2052 1191.5293 -343.274686 2726.333 -1155.75082 3538.809
## Oct 2052 1183.7155 -352.973221 2720.404 -1166.44708 3533.878
## Nov 2052 1190.9688 -347.602283 2729.540 -1162.07264 3544.010
## Dec 2052 1193.6636 -346.787686 2734.115 -1162.25333 3549.580
## Jan 2053 1218.8839 -323.445203 2761.213 -1139.90491 3577.673
## Feb 2053 1180.3071 -363.897524 2724.512 -1181.35009 3541.964
## Mar 2053 1225.4670 -320.610954 2771.545 -1139.05518 3589.989
## Apr 2053 1227.5480 -320.400967 2775.497 -1139.83565 3594.932
## May 2053 1228.6543 -321.163465 2778.472 -1141.58741 3598.896
## Jun 2053 1224.7146 -326.969597 2776.399 -1148.38161 3597.811
## Jul 2053 1222.7570 -330.791503 2776.305 -1153.19040 3598.704
## Aug 2053 1182.2781 -373.132380 2737.689 -1196.51697 3561.073
## Sep 2053 1200.4154 -356.854941 2757.686 -1181.22405 3582.055
## Oct 2053 1190.2700 -368.857863 2749.398 -1194.21032 3574.750
## Nov 2053 1195.8689 -365.114342 2756.852 -1191.44898 3583.187
## Dec 2053 1209.7791 -353.057350 2772.616 -1180.37300 3599.931
## Jan 2054 1221.9633 -342.724093 2786.651 -1171.01959 3614.946
## Feb 2054 1211.1274 -355.408783 2777.664 -1184.68297 3606.938
## Mar 2054 1225.0689 -343.313971 2793.452 -1173.56569 3623.703
## Apr 2054 1242.4160 -327.811275 2812.643 -1159.03938 3643.871
## May 2054 1237.3060 -334.763569 2809.376 -1166.96692 3641.579
## Jun 2054 1244.3167 -329.592951 2818.226 -1162.77040 3651.404
## Jul 2054 1255.5171 -320.230583 2831.265 -1154.38099 3665.415
## Aug 2054 1233.9728 -343.610695 2811.556 -1178.73294 3646.678
## Sep 2054 1255.2932 -324.123995 2834.710 -1160.21694 3670.803
## Oct 2054 1250.9858 -330.262934 2832.235 -1167.32545 3669.297
## Nov 2054 1259.2738 -323.804348 2842.352 -1161.83531 3680.383
## Dec 2054 1257.2049 -327.700647 2842.110 -1166.69895 3681.109
## Jan 2055 1254.2247 -332.505998 2840.955 -1172.47052 3680.920
## Feb 2055 1271.9145 -316.639418 2860.468 -1157.56905 3701.398
## Mar 2055 1262.8071 -327.567799 2853.182 -1169.46143 3695.076
## Apr 2055 1280.4315 -311.762391 2872.625 -1154.61892 3715.482
## May 2055 1291.1454 -302.865362 2885.156 -1146.68370 3728.975
## Jun 2055 1277.8058 -318.019807 2873.631 -1162.79885 3718.410
## Jul 2055 1277.0022 -320.636129 2874.641 -1166.37479 3720.379
## Aug 2055 1261.8177 -337.631392 2861.267 -1184.32858 3707.964
## Sep 2055 1284.2127 -317.045017 2885.470 -1164.69965 3733.125
## Oct 2055 1295.2635 -307.800806 2898.328 -1156.41180 3746.939
## Nov 2055 1251.1648 -353.704118 2856.034 -1203.27040 3705.600
## Dec 2055 1269.6280 -337.043475 2876.299 -1187.56397 3726.820
## Jan 2056 1241.8317 -366.640344 2850.304 -1218.11398 3701.777
## Feb 2056 1232.5077 -377.762828 2842.778 -1230.18855 3695.204
## Mar 2056 1216.7642 -395.302811 2828.831 -1248.67955 3682.208
## Apr 2056 1221.5990 -392.262569 2835.461 -1246.58926 3689.787
## May 2056 1231.4524 -384.201665 2847.106 -1239.47726 3702.382
## Jun 2056 1231.2532 -386.191446 2848.698 -1242.41490 3704.921
## Jul 2056 1214.3039 -404.929261 2833.537 -1262.09951 3690.707
## Aug 2056 1228.0171 -393.002684 2849.037 -1251.11870 3707.153
## Sep 2056 1233.3260 -389.478377 2856.130 -1248.53911 3715.191
## Oct 2056 1221.6036 -402.983416 2846.191 -1262.98783 3706.195
## Nov 2056 1221.5606 -404.807061 2847.928 -1265.75412 3708.875
## Dec 2056 1240.6724 -387.474031 2868.819 -1249.36270 3730.708
## Jan 2057 1226.0587 -403.864562 2855.982 -1266.69382 3718.811
## Feb 2057 1242.2364 -389.461709 2873.935 -1253.23053 3737.703
## Mar 2057 1246.0896 -387.381507 2879.561 -1252.08887 3744.268
## Apr 2057 1239.0875 -396.154657 2874.330 -1261.79954 3739.974
## May 2057 1236.3817 -400.629504 2873.393 -1267.21090 3739.974
## Jun 2057 1242.4706 -396.307863 2881.249 -1263.82476 3748.766
## Jul 2057 1261.5798 -378.963945 2902.123 -1247.41533 3770.575
## Aug 2057 1261.5193 -380.787789 2903.826 -1250.17266 3773.211
## Sep 2057 1234.0689 -409.999691 2878.138 -1280.31705 3748.455
## Oct 2057 1260.5024 -385.325872 2906.331 -1256.57472 3777.579
## Nov 2057 1291.7501 -355.835869 2939.336 -1228.01521 3811.515
## Dec 2057 1247.0393 -402.302546 2896.381 -1275.41139 3769.490
## Jan 2058 1245.1328 -405.963031 2896.229 -1280.00038 3770.266
## Feb 2058 1245.3555 -407.492535 2898.203 -1282.45742 3773.168
## Mar 2058 1248.3096 -406.288708 2902.908 -1282.18013 3778.799
## Apr 2058 1257.8459 -398.500792 2914.193 -1275.31778 3791.010
## May 2058 1240.1463 -417.946992 2898.240 -1295.68857 3775.981
## Jun 2058 1252.8264 -407.011708 2912.664 -1285.67691 3791.330
## Jul 2058 1232.8600 -428.720976 2894.441 -1308.30882 3774.029
## Aug 2058 1272.4817 -390.840386 2935.804 -1271.34991 3816.313
## Sep 2058 1266.5384 -398.522936 2931.600 -1279.95318 3813.030
## Oct 2058 1251.4424 -415.356460 2918.241 -1297.70646 3800.591
## Nov 2058 1264.6493 -403.885161 2933.184 -1287.15396 3816.453
## Dec 2058 1274.4523 -395.816029 2944.721 -1280.00267 3828.907
## Jan 2059 1276.0439 -395.956443 2948.044 -1281.05997 3833.148
## Feb 2059 1280.0112 -393.719424 2953.742 -1279.73890 3839.761
## Mar 2059 1266.1040 -409.355100 2941.563 -1296.28957 3828.498
## Apr 2059 1275.6557 -401.530134 2952.841 -1289.37866 3840.690
## May 2059 1279.5444 -399.366309 2958.455 -1288.12795 3847.217
## Jun 2059 1288.9662 -391.667610 2969.600 -1281.34143 3859.274
## Jul 2059 1223.8738 -458.481467 2906.229 -1349.06652 3796.814
## Aug 2059 1231.3256 -452.749230 2915.400 -1344.24460 3806.896
## Sep 2059 1252.1827 -433.609962 2937.975 -1326.01471 3830.380
## Oct 2059 1236.9405 -450.568325 2924.449 -1343.88153 3817.763
## Nov 2059 1219.8688 -469.354442 2909.092 -1363.57518 3803.313
## Dec 2059 1261.4730 -429.462856 2952.409 -1324.59021 3847.536
## Jan 2060 1246.3763 -446.270407 2939.023 -1342.30346 3835.056
## Feb 2060 1245.5895 -448.766426 2939.945 -1345.70426 3836.883
## Mar 2060 1233.7678 -462.295548 2929.831 -1360.13725 3827.673
## Apr 2060 1261.0788 -436.690236 2958.848 -1335.43490 3857.593
## May 2060 1269.9521 -429.521019 2969.425 -1329.16774 3869.072
## Jun 2060 1235.1659 -466.009500 2936.341 -1366.55737 3836.889
## Jul 2060 1257.1235 -445.752493 2960.000 -1347.20061 3861.448
## Aug 2060 1242.6370 -461.937995 2947.212 -1364.28547 3849.559
## Sep 2060 1249.6720 -456.600204 2955.944 -1359.84613 3859.190
## Oct 2060 1211.6245 -496.343165 2919.592 -1400.48666 3823.736
## Nov 2060 1214.1275 -495.534054 2923.789 -1400.57422 3828.829
## Dec 2060 1223.3668 -487.986923 2934.721 -1393.92288 3840.657
## Jan 2061 1208.0461 -504.998133 2921.090 -1411.82899 3827.921
## Feb 2061 1212.6192 -502.113937 2927.352 -1409.83881 3835.077
## Mar 2061 1215.2451 -501.175232 2931.665 -1409.79324 3840.283
## Apr 2061 1206.6905 -511.415259 2924.796 -1420.92553 3834.307
## May 2061 1185.1474 -534.642303 2904.937 -1445.04397 3815.339
## Jun 2061 1208.4147 -513.057197 2929.887 -1424.34938 3841.179
## Jul 2061 1289.0701 -434.082379 3012.223 -1346.26420 3924.404Maximum Likelihood Estimation in ARIMA models
R estimates the ARIMA model using maximum likelihood estimation (MLE). This approch finds the parameter values that maximize the probability of obtaining the observed data.
R will report the value of the log likelihood of the data; that is, the logarithm of the probability of the observed data coming from the estimated model. For given values of p,d and q, R will try to maximise the log likelihood when finding parameter estimates.
Below I have created a model that has the lowest value of AIC. I have looped over several combinations of p,d and q and store the fitted model of ARIMA(p,d,q). If the current AIC value is less than the previously generated AIC, then the current AIC is the final AIC and select the order. After terminating the loop, I have stored the order of the ARIMA model in final.order and stored ARIMA(p,d,q) fitted model in final.arima
Upon termination of the loop we have the order of the ARIMA model stored in final.order and the ARIMA(p,d,q) fit itself stored as final.arma:
##
## Call:
## arima(x = time_ser[, 1], order = house_final.order)
##
## Coefficients:
## ar1 ar2 ar3 ar4 ma1 ma2 ma3 ma4
## 0.0022 0.5383 -0.5267 -0.1795 -1.3846 -0.1697 1.3837 -0.8293
## s.e. 0.1021 0.1199 0.0883 0.0572 0.0899 0.0708 0.0947 0.0613
##
## sigma^2 estimated as 10340: log likelihood = -3079.64, aic = 6177.28##
## Call:
## arima(x = time_ser[, 1], order = c(4, 2, 4))
##
## Coefficients:
## ar1 ar2 ar3 ar4 ma1 ma2 ma3 ma4
## 0.0022 0.5383 -0.5267 -0.1795 -1.3846 -0.1697 1.3837 -0.8293
## s.e. 0.1021 0.1199 0.0883 0.0572 0.0899 0.0708 0.0947 0.0613
##
## sigma^2 estimated as 10340: log likelihood = -3079.64, aic = 6177.28
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -2.584756 101.4876 76.39623 -0.4136831 5.737581 0.914796
## ACF1
## Training set -0.006148856After simulating the AIRMA model of order ARIMA(4,2,4), the model is:
y_t = c + 0.0022 y_(t-1) + 0.5383y_(t-2) + -0.5267 y_(t-3) + -0.1795 y_(t-4) + -1.3846ε_(t-1) + -0.1697ε_(t-2) + 1.3837ε_(t-3)+ -0.8293 ε_(t43)
where ε_t is white noise with standard deviation of sqrt(10340) = 101.6858.
The ARCH-LM test with q lags checks for the presence of ARCH effects at lags 1 to q. It tests if the coefficients α_1,…. α_q in the equation below:
x^2_t = α_0 + α_1 * x^2_(t-1) +….+ α_q * x^2_(t-q) + ϵ_t
## ARCH heteroscedasticity test for residuals
## alternative: heteroscedastic
##
## Portmanteau-Q test:
## order PQ p.value
## [1,] 4 44.8 4.35e-09
## [2,] 8 46.0 2.35e-07
## [3,] 12 107.1 0.00e+00
## [4,] 16 138.6 0.00e+00
## [5,] 20 143.9 0.00e+00
## [6,] 24 165.1 0.00e+00
## Lagrange-Multiplier test:
## order LM p.value
## [1,] 4 238.4 0.00e+00
## [2,] 8 117.8 0.00e+00
## [3,] 12 73.5 2.65e-11
## [4,] 16 41.0 3.25e-04
## [5,] 20 31.1 3.95e-02
## [6,] 24 22.6 4.82e-01
As the p-value is very small, we reject the null hypothesis and conclude that ARCH effects are present at lags 1 and 2 jointly. ARCH effects are also present at higher lag orders, implying that the data is conditionally heteroskedastic.
Generalized Autoregressive Conditional Heteroskedasticity (GARCH)
Generalized Autoregressive Conditional Heteroskedastic, or GARCH models are useful to analyse and forecast volatility in a time series data. Univariate GARCH(1,1) helps in modeling volality and its clustering.
Financial time series possess the property of volatility clustering wherein the volatility of the variable changes over time. Technically, this behavior is called conditional heteroskedasticity. Because ARMA models don’t consider volatility clustering i.e. they are not conditionally heteroskedastic, so we need to use ARCH and GARCH models for predictions.
Such models include the Autogressive Conditional Heteroskedastic (ARCH) model and Generalised Autogressive Conditional Heteroskedastic (GARCH) model. Different forms of volatility such as sell-offs during a financial crises, can cause serially correlated heteroskedasticity. Thus, the time_ser data is conditionally heteroskedastic.
Maximum likelihood estimates most GARCH models, such as measuring relative loss or profit from trading stocks in a day. If x_t is the value of housing starts on t, then r_t=[x_t − x_(t−1)]/x_(t−1) is called the return. We observe large volatility around the 2008 financial crisis and returns that are mostly noise noise with short periods of large variability.
Here, we test if the returns of housing starts are autocorrelated
## ChiSq DF pvalue
## [1,] 55.53981 5 1.010765e-10
## [2,] 60.73615 10 2.629113e-09
## [3,] 86.60523 20 2.889828e-10
## attr(,"method")
## [1] "LjungBox"We reject the hypothesis that the series is independently and identically distributed from the Ljung-Box test.The series is not white noise, hence autocorrelated.
Below I have plotted the ACF of the returns of housing starts. There are two bounds plotted on the graph. The straight red line represents the standard bounds under the strong white noise assumption. The second line is under the hypothesis that the process is GARCH.
From the plot above, several autocorrelations seem significant under hypothesis of both iid and GARCH process.
Now, I have fit a GARCH-type model which assumes the null hypothesis that the returns are GARCH.
## h Q pval
## [1,] 5 40.57540 1.143075e-07
## [2,] 10 42.76489 5.478140e-06
## [3,] 15 52.29641 5.045273e-06The low p-values give reason to reject the hypothesis that the returns are a GARCH white noise process. So, we should do ARMA modelling.
We have fit GARCH model(s), starting with a GARCH(1,1) model with Gaussian innovations.GARCH(1,1) considers a single autoregressive and a moving average lag. The model is:
ϵ_t = σ_t * w_t σ^2 = α_0 + α_1 * ϵ^2_(t−1) + β_1 * σ^2_(t−1)
Note that alpha_1 + beta_1 < 0, otherwise the series will become unstable.
The persistence of a GARCH model signifies the rate at which large volatilities decay after a shock. The key statistic in GARCH(1,1) is the sum of two parameters: alpha1 and beta1.
Ideally, alpha_1 + beta_1 < 1. If, alpha_1 + beta_1 > 1, then the volatility predictions are explosive. If, alpha_1 + beta_1 = 1, then the model has exponential decay.
In the output from garchFit, the normalized log-likelihood is the loglikelihood divided by n. The AIC and BIC values have also been normalized by dividing by n,
##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~arma(1, 0) + garch(1, 1), data = hous_st_return,
## cond.dist = "norm")
##
## Mean and Variance Equation:
## data ~ arma(1, 0) + garch(1, 1)
## <environment: 0x562be5588ea0>
## [data = hous_st_return]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu ar1 omega alpha1 beta1
## 4.1271e-06 -3.6658e-01 6.5196e-05 6.8848e-02 9.2174e-01
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 4.127e-06 2.994e-03 0.001 0.99890
## ar1 -3.666e-01 4.253e-02 -8.619 < 2e-16 ***
## omega 6.520e-05 5.193e-05 1.256 0.20929
## alpha1 6.885e-02 2.182e-02 3.155 0.00161 **
## beta1 9.217e-01 2.286e-02 40.319 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 599.6187 normalized: 1.178033
##
## Description:
## Mon Apr 1 22:22:05 2019 by user:
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 18.01514 0.0001224791
## Shapiro-Wilk Test R W 0.9891823 0.0008269949
## Ljung-Box Test R Q(10) 16.03317 0.09868677
## Ljung-Box Test R Q(15) 27.71855 0.02339867
## Ljung-Box Test R Q(20) 34.76176 0.02141044
## Ljung-Box Test R^2 Q(10) 26.98303 0.002620484
## Ljung-Box Test R^2 Q(15) 43.25259 0.0001438445
## Ljung-Box Test R^2 Q(20) 46.94882 0.0005962632
## LM Arch Test R TR^2 26.92906 0.007910933
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -2.336419 -2.294843 -2.336610 -2.320117The diagnostics imply that the standardised residuals and their squares are IID and that the model accomodates ARCH effects.
H_0: white noise innovation process is Gaussian
Their distribution is Gaussian only from the p-value for Ljung-Box Test which is 0.921266. From all other tests of normality, we reject the null hypothesis as the p-values are very low.
The qq-plot of the standardised residuals, suggests that the fitted standardised skew-t conditional distribution is decent.
Since, ARIMA linearly models the data, the forecast width is constant as the model does not incorporate new information or recent changes.To model non-linearity or a cluster of volatility, we have to use ARCH/GARCH methods as they reflect more recent fluctuations in the series. The ACF and PACF of residuals can confirm if the residuals can be predicted if they are not white noise. Residuals of strict white noise series are i.i.d normally distributed with zero mean. Moreover, the PACF and ACF of squared residuals have no significant lags. Finally, we cannot predict a strict white noise series, either linearly or non-linearly. Below, the residuals and squared residuals of ARIMA(4,2,4) model show a cluster of volatility as shown from the ACF plots.
##
## Box-Ljung test
##
## data: resid
## X-squared = 6.1032, df = 10, p-value = 0.8065H_o: no autocorrelation
We fail to reject the null hypothesis that the residuals of ARIMA(4,2,4) is not serially correlated i.e. We conclude that the residuals of ARIMA(4,2,4) follow a white noise process.
##
## Box-Ljung test
##
## data: resid^2
## X-squared = 63.898, df = 10, p-value = 6.583e-10H_o: no autocorrelation
We fail to reject the null hypothesis that the squared residuals of ARIMA(4,2,4) is not serially correlated. We conclude that the squared residuals of ARIMA(4,2,4) do not follow a white noise process and are autocorrelated. So, the time series exhibits conditional heteroskedasticity. Now, I have fit a GARCH(1,1) model.
##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~arma(4, 4) + garch(1, 1), data = time_ser_diff[,
## 1])
##
## Mean and Variance Equation:
## data ~ arma(4, 4) + garch(1, 1)
## <environment: 0x562be9fc7a90>
## [data = time_ser_diff[, 1]]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu ar1 ar2 ar3 ar4
## 16.053017 0.471082 0.579245 0.662428 -0.724449
## ma1 ma2 ma3 ma4 omega
## 0.083284 -0.252425 -0.649781 0.336848 2640.659451
## alpha1 beta1
## 0.424977 0.367700
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 16.05302 11.47482 1.399 0.161820
## ar1 0.47108 0.15926 2.958 0.003096 **
## ar2 0.57925 0.05429 10.670 < 2e-16 ***
## ar3 0.66243 0.05118 12.942 < 2e-16 ***
## ar4 -0.72445 0.14914 -4.858 1.19e-06 ***
## ma1 0.08328 0.15740 0.529 0.596715
## ma2 -0.25242 0.14761 -1.710 0.087245 .
## ma3 -0.64978 0.10502 -6.187 6.12e-10 ***
## ma4 0.33685 0.05913 5.697 1.22e-08 ***
## omega 2640.65945 773.37713 3.414 0.000639 ***
## alpha1 0.42498 0.09702 4.380 1.19e-05 ***
## beta1 0.36770 0.10534 3.491 0.000482 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -3050.358 normalized: -5.96939
##
## Description:
## Mon Apr 1 22:22:07 2019 by user:
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 10.53292 0.005161848
## Shapiro-Wilk Test R W 0.9928411 0.01531213
## Ljung-Box Test R Q(10) 4.61827 0.9151778
## Ljung-Box Test R Q(15) 15.15126 0.4405772
## Ljung-Box Test R Q(20) 19.6149 0.4822394
## Ljung-Box Test R^2 Q(10) 10.52377 0.3958027
## Ljung-Box Test R^2 Q(15) 39.47513 0.0005439962
## Ljung-Box Test R^2 Q(20) 40.64349 0.004138153
## LM Arch Test R TR^2 20.30752 0.06148775
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## 11.98575 12.08523 11.98468 12.02475