ASX Series Analysis
Bhavya Dinesh (s3980321)
2023-08-08
Introduction
In this analysis, we explore the ASX dataset which features monthly averages of ASX all Oridinaries (Ords) Price Index, Gold Price (AUD), Crude Oil(Brent,USD/bbl) and Copper(USD/tonne) starting from January 2004.
Aim of this analysis:
- Determine the presence of nonstationarity within the dataset.
- Investigate the impact of individual time series components.
- Identify the most suitable distributed lag model for accurate ASX Price Index prediction
ASX_data <- read_csv("ASX_data.csv")## Rows: 161 Columns: 4
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## dbl (2): ASX price, Crude Oil (Brent)_USD/bbl
## num (2): Gold price, Copper_USD/tonne
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.head(ASX_data)## # A tibble: 6 × 4
## `ASX price` `Gold price` `Crude Oil (Brent)_USD/bbl` `Copper_USD/tonne`
## <dbl> <dbl> <dbl> <dbl>
## 1 2935. 612. 31.3 1650
## 2 2778. 603. 32.6 1682
## 3 2849. 566. 30.3 1656
## 4 2971. 539. 25.0 1588
## 5 2980. 549. 25.8 1651
## 6 3000. 536. 27.6 1685Descriptive Statistics
summary(ASX_data)## ASX price Gold price Crude Oil (Brent)_USD/bbl Copper_USD/tonne
## Min. :2778 Min. : 520.9 Min. : 25.02 Min. :1588
## 1st Qu.:4325 1st Qu.: 825.9 1st Qu.: 47.23 1st Qu.:4642
## Median :4929 Median :1363.6 Median : 70.80 Median :6650
## Mean :4813 Mean :1202.0 Mean : 73.63 Mean :5989
## 3rd Qu.:5448 3rd Qu.:1546.8 3rd Qu.:106.98 3rd Qu.:7623
## Max. :6779 Max. :1776.3 Max. :133.90 Max. :9881Time series Plots
#converting ASX dataframes to monthly time series objects
asx.ts <- ts(ASX_data$`ASX price`, start = c(2004,1) ,frequency = 12)
gold.ts <- ts(ASX_data$`Gold price`, start = c(2004,1) ,frequency = 12)
crudeoil.ts <- ts(ASX_data$`Crude Oil (Brent)_USD/bbl`, start = c(2004,1), frequency = 12)
copper.ts <- ts(ASX_data$`Copper_USD/tonne`, start = c(2004,1) ,frequency = 12)ASX Series Plot
plot(asx.ts,type='o', pch=20, ylab="Ords",xlab="Year",main ="ASX (ORDS) Price Index Time Series Plot")
The ASX series plot shows an upward trend for the given time period. There are no signs of seasonality and changing variance. Moreover, an intervention point is present between 2007 and 2009. The succeeding points indicate the presence of an autoregressive pattern.
Gold Series Plot
plot(gold.ts, type='o', pch=20 ,ylab="Aud",xlab="Year",main ="Gold price Time Series Plot")
The gold series plot shows an upward trend for the given time period. There are no signs of seasonality and changing variance. Moreover, no intervention point can be observed. The succeeding points indicate the presence of an autoregressive pattern.
Crude Oil Series Plot
plot(crudeoil.ts,type='o', pch=20, ylab="USD/bbl",xlab="Year",main ="Crude oil price Time Series Plot")
The crude series plot shows no obvious trend for the given time period,
no signs of seasonality and changing variance. Moreover, an intervention
point is present around 2008. The succeeding points indicate the
presence of an autoregressive pattern.
Copper Series Plot
plot(copper.ts,type='o', pch=20, ylab="USD/tonne",xlab="Year",main ="Copper price Time Series Plot")
The copper series plot shows no apparent trend for the given time period. There are no signs of seasonality and changing variance. Moreover, an intervention point is present between 2007 and 2009. The succeeding points indicate the presence of an autoregressive pattern.
ASX All series plot
To gain an insight into the relationship between our independent time series (Gold, Crude Oil, Copper) and the dependent series (All Ordinaries ASX), we present them together in a single plot which is appropriately scaled and centered.
data.ts = ts(ASX_data, start = 2004, frequency = 12) #converting data to time series object
data.scale=scale(data.ts) #scaling data
plot(data.scale, plot.type="s", col=c("black", "gold","skyblue","lightgreen"),
main = "ASX All Series", xlab="Year", ylab="Scaled Data")
legend ("topleft", lty =1, text.width=1.5, col=c("black", "gold","skyblue","lightgreen"),
c('All Ords', 'Gold', 'Oil', 'Copper'))
The ASX all series plot show an upward trend until the intervention point around 2008. Only the gold price shows a stable upward trend. Copper and Oil price follow similar patterns hence suggesting a correlation between both prices.
Stationarity Analysis
ACF and PACF plot
par(mfrow=c(1,2))
acf(asx.ts, main = "Sample ACF for ASX series", )
pacf(asx.ts, main = "Sample PACF for ASX series")
par(mfrow=c(1,2))
acf(gold.ts, main = "Sample ACF for GOLD series", )
pacf(gold.ts, main = "Sample PACF for GOLD series")
par(mfrow=c(1,2))
par(mar=c(4,4,4,4))
acf(crudeoil.ts, main = "Sample ACF for Crude Oil series", )
pacf(crudeoil.ts, main = "Sample PACF for Crude Oil series")
par(mfrow=c(1,2))
par(mar=c(4,4,4,4))
acf(copper.ts, main = "Sample ACF for Copper Oil series" )
pacf(copper.ts, main = "Sample PACF for Copper Oil series")
With respect to all ACF plots.We observe positive values which gradually
decrease as the lags increase. This pattern strongly suggests the
presence of a trend in the series. Additionally, across all PACF plots,
the significance of the first lag suggests the presence of
non-stationarity within the series.
ADF test
To determine if the series is stationary we perform Augmented Dickey-Fuller test (ADF) which is defined as the following hypothesis.
H0: The time series is non-stationary. In other words, it has some time-dependent structure and does not have constant variance over time.
HA: The time series is stationary.
k = ar(asx.ts)$order # length of lag for ADF
adf.test(asx.ts,k=k)##
## Augmented Dickey-Fuller Test
##
## data: asx.ts
## Dickey-Fuller = -2.442, Lag order = 2, p-value = 0.392
## alternative hypothesis: stationaryk = ar(gold.ts)$order
adf.test(gold.ts,k=k)##
## Augmented Dickey-Fuller Test
##
## data: gold.ts
## Dickey-Fuller = -2.2824, Lag order = 1, p-value = 0.4585
## alternative hypothesis: stationaryk = ar(crudeoil.ts)$order
adf.test(crudeoil.ts,k=k)##
## Augmented Dickey-Fuller Test
##
## data: crudeoil.ts
## Dickey-Fuller = -2.0703, Lag order = 3, p-value = 0.547
## alternative hypothesis: stationaryk = ar(copper.ts)$order
adf.test(copper.ts,k=k)##
## Augmented Dickey-Fuller Test
##
## data: copper.ts
## Dickey-Fuller = -2.3847, Lag order = 2, p-value = 0.4159
## alternative hypothesis: stationaryThe ADF test for all series have a P-value > 0.05 therefore we fail to reject the H0 (null hypothesis) and conclude all series are non-stationary at 5% level.
Decomposition
STL and x12 decomposition is used to examine the impact of trend and seasonality components within the series.
asx.stl <- stl(asx.ts, t.window=15, s.window = "periodic", robust=TRUE);
plot(asx.stl, main ="STL Decomposition of ASX series")
asx.x12 <- x12(asx.ts)
plot(asx.x12, sa= TRUE, trend= TRUE, main= "X12 Decomposition of ASX series", forecast=TRUE)
The remainder component in the STL plot indicates significant noise near
the intervention point. ASX time series plot and ACF plot showed no
signs of seasonality therefore the STL seasonal plot has no meaningful
insights. X12 decomposition plot shows seasonally adjusted series is
close to the original series.
gold.stl <- stl(gold.ts, t.window=15, s.window = "periodic", robust=TRUE);
plot(asx.stl, main ="STL Decomposition of Gold series")
gold.x12 <- x12(gold.ts)
plot(asx.x12, sa= TRUE, trend= TRUE, main= "X12 Decomposition of Gold series", forecast=TRUE)
There is significant fluctuations in the remainder component of the STL
Gold series plot which indicates there are unknown factors influencing
the series which cannot be explained by the trend and seasonal
components. Additionally, gold time series plot and ACF plot showed no
signs of seasonality therefore the STL seasonal plot has no meaningful
insights and X12 decomposition plot shows seasonally adjusted series is
close to the original series.
crudeoil.stl <- stl(crudeoil.ts, t.window=15, s.window = "periodic", robust=TRUE);
plot(asx.stl, main ="STL Decomposition of CrudeOil series")
asx.x12 <- x12(crudeoil.ts)
plot(asx.x12, sa= TRUE, trend= TRUE, main= "X12 Decomposition of CrudeOil series", forecast=TRUE)
There is significant fluctuations in the remainder component near the intervention point of the STL Crude oil series plot which indicates there are unknown factors influencing the series which cannot be explained by the trend and seasonal components. Additionally, Crude oil time series plot and ACF plot showed no signs of seasonality therefore the STL seasonal plot has no meaningful insights and X12 decomposition plot shows seasonally adjusted series is close to the original series.
copper.stl <- stl(copper.ts, t.window=15, s.window = "periodic", robust=TRUE);
plot(asx.stl, main ="STL Decomposition of Copper series")
asx.x12 <- x12(copper.ts)
plot(asx.x12, sa= TRUE, trend= TRUE, main= "X12 Decomposition of Copper series",forecast=TRUE)
There is significant fluctuations in the remainder component near the intervention period of the STL Copper series plot suggests there are unknown factors influencing the series which cannot be explained by the trend and seasonal components. Additionally, Copper time series plot and ACF plot showed no signs of seasonality therefore the STL seasonal plot has no meaningful insights and X12 decomposition plot shows seasonally adjusted series is close to the original series.
Correlation Matrix
colnames(data.ts) <- c('ASXPrice', 'GoldPrice', 'CrudeOilPrice', 'CopperPrice') #renaming columns
corr <- cor(data.ts) # Correlation
head(corr)## ASXPrice GoldPrice CrudeOilPrice CopperPrice
## ASXPrice 1.0000000 0.3431908 0.3290338 0.5617864
## GoldPrice 0.3431908 1.0000000 0.4366382 0.5364213
## CrudeOilPrice 0.3290338 0.4366382 1.0000000 0.8664296
## CopperPrice 0.5617864 0.5364213 0.8664296 1.0000000colors <- dmat.color(corr) #coloring according to correlation strength
# (pink for high and yellow for low correlation )
order <- order.single(corr) # rearranges matrix to keep variables
#with high correlation close to each other.
cpairs(data.ts, order, panel.colors = colors, gap = 0.5,
main = "Correlation Scatter Plot Sorted")
The correlation matrix and scatter plot both indicate a strong positive correlation between crude oil and copper price (r=0.866). Additionally, there is a significant correlation between ASX price and copper price (r=0.566). Due to high correlation between crude oil and copper, we do not use both of them as independent variables simultaneously during the modelling process.
Modelling
We perform 4 different types of model to determine the most suited model for the ASX price index. In Finite DLM and Autoregressive DLM we use copper and gold price series as our predictors. Polynomial DLM and koyck transformation we use copper price series as our predictor since it has the highest correlation with ASX price index.
Finite DLM
for (i in 1:10){
model1.1 = dlm(x = as.vector(gold.ts)+as.vector(copper.ts), y = as.vector(asx.ts), q = i)
cat("q = ", i, "AIC = ", AIC(model1.1$model), "BIC = ", BIC(model1.1$model), "\n")
}## q = 1 AIC = 2574.318 BIC = 2586.619
## q = 2 AIC = 2559.216 BIC = 2574.56
## q = 3 AIC = 2543.874 BIC = 2562.25
## q = 4 AIC = 2528.381 BIC = 2549.774
## q = 5 AIC = 2512.536 BIC = 2536.935
## q = 6 AIC = 2496.908 BIC = 2524.299
## q = 7 AIC = 2480.908 BIC = 2511.277
## q = 8 AIC = 2465.188 BIC = 2498.522
## q = 9 AIC = 2449.696 BIC = 2485.983
## q = 10 AIC = 2435.031 BIC = 2474.256The lag length q=10 denotes the lowest AIC and BIC values.
model1 = dlm(x = as.vector(gold.ts)+as.vector(copper.ts), y = as.vector(asx.ts), q =10)
summary(model1)##
## Call:
## lm(formula = model.formula, data = design)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1207.74 -647.59 44.66 592.64 1504.89
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3875.39268 229.14655 16.912 <2e-16 ***
## x.t 0.14749 0.13472 1.095 0.275
## x.1 0.02523 0.21748 0.116 0.908
## x.2 0.04443 0.21875 0.203 0.839
## x.3 0.03152 0.21540 0.146 0.884
## x.4 0.01588 0.21505 0.074 0.941
## x.5 -0.04433 0.21711 -0.204 0.839
## x.6 0.02544 0.21507 0.118 0.906
## x.7 0.00347 0.21645 0.016 0.987
## x.8 -0.01022 0.22098 -0.046 0.963
## x.9 -0.07916 0.21945 -0.361 0.719
## x.10 -0.02261 0.13311 -0.170 0.865
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 734.6 on 139 degrees of freedom
## Multiple R-squared: 0.1991, Adjusted R-squared: 0.1357
## F-statistic: 3.142 on 11 and 139 DF, p-value: 0.0008096
##
## AIC and BIC values for the model:
## AIC BIC
## 1 2435.031 2474.256checkresiduals(model1$model)
##
## Breusch-Godfrey test for serial correlation of order up to 15
##
## data: Residuals
## LM test = 139.03, df = 15, p-value < 2.2e-16From the Model 1 summary and residual diagnostics we observe:
No lags are significant at 5% level.
Adjust R-squared is very low at 0.1357 which means it only explains 13.57% of the variability.
The p-value: 0.0008096 is < 0.05 therefore the model is significant.
Residual plot shows a trend in the series.
ACF plot indicates presence of a serial correlation in the residuals.
Histogram is not normally distributed.
Breusch-Godfrey test p-value < 0.05 therefore there is significant autocorrelation remaining in the residuals.
VIF.model1 = vif(model1$model)
VIF.model1## x.t x.1 x.2 x.3 x.4 x.5 x.6 x.7
## 21.56171 58.40449 61.40224 61.79140 63.86995 67.40979 68.40573 71.60574
## x.8 x.9 x.10
## 76.87451 77.94537 29.46896VIF.model1 > 10## x.t x.1 x.2 x.3 x.4 x.5 x.6 x.7 x.8 x.9 x.10
## TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE- The value of VIF for model 1 is greater than 10 therefore the effect of multicollinearity is high.
Polynomial DLM
model2 = polyDlm(x = as.vector(copper.ts), y = as.vector(asx.ts), q = 10, k = 2, show.beta = TRUE)## Estimates and t-tests for beta coefficients:
## Estimate Std. Error t value P(>|t|)
## beta.0 0.09760 0.03120 3.130 2.14e-03
## beta.1 0.07150 0.01650 4.330 2.84e-05
## beta.2 0.04850 0.00916 5.300 4.44e-07
## beta.3 0.02860 0.01180 2.420 1.69e-02
## beta.4 0.01170 0.01580 0.741 4.60e-01
## beta.5 -0.00205 0.01730 -0.118 9.06e-01
## beta.6 -0.01270 0.01580 -0.806 4.22e-01
## beta.7 -0.02040 0.01170 -1.750 8.31e-02
## beta.8 -0.02490 0.00830 -3.000 3.18e-03
## beta.9 -0.02640 0.01530 -1.720 8.72e-02
## beta.10 -0.02480 0.02990 -0.829 4.09e-01summary(model2)##
## Call:
## "Y ~ (Intercept) + X.t"
##
## Residuals:
## Min 1Q Median 3Q Max
## -1180.62 -642.64 -40.32 616.63 1484.83
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.997e+03 2.088e+02 19.144 < 2e-16 ***
## z.t0 9.756e-02 3.119e-02 3.128 0.00212 **
## z.t1 -2.761e-02 1.802e-02 -1.532 0.12761
## z.t2 1.537e-03 1.772e-03 0.868 0.38706
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 718.2 on 147 degrees of freedom
## Multiple R-squared: 0.1906, Adjusted R-squared: 0.1741
## F-statistic: 11.54 on 3 and 147 DF, p-value: 7.739e-07checkresiduals(model2$model)
##
## Breusch-Godfrey test for serial correlation of order up to 10
##
## data: Residuals
## LM test = 138.14, df = 10, p-value < 2.2e-16From the model 2 summary and residual diagnostics we observe:
No lags are significant at 5% level
Adjust R-squared is very low at 0.1741 which means it only explains 17.41% of the variability.
p-value:7.739e-07 < 0.05 therefore the model is significant.
Residual plot shows a trend in the series.
ACF plot indicates presence of a serial correlation in the residuals
Histogram is not normally distributed
Breusch-Godfrey test p-value is < 0.05 therefore there is significant autocorrelation remaining in the residuals.
VIF.model2 = vif(model2$model)
VIF.model2## z.t0 z.t1 z.t2
## 121.4442 1131.9986 565.1003VIF.model2 > 10## z.t0 z.t1 z.t2
## TRUE TRUE TRUE- The value of VIF for model 2 is greater than 10 therefore the effect of multicollinearity is high.
Koyck DLM
model3 = koyckDlm(x = as.vector(copper.ts), y = as.vector(asx.ts))
summary(model3)##
## Call:
## "Y ~ (Intercept) + Y.1 + X.t"
##
## Residuals:
## Min 1Q Median 3Q Max
## -689.64 -108.62 12.78 140.20 771.79
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 189.368812 87.644648 2.161 0.0322 *
## Y.1 0.971621 0.021895 44.376 <2e-16 ***
## X.t -0.005864 0.009517 -0.616 0.5387
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 201.9 on 157 degrees of freedom
## Multiple R-Squared: 0.9485, Adjusted R-squared: 0.9479
## Wald test: 1448 on 2 and 157 DF, p-value: < 2.2e-16
##
## Diagnostic tests:
## NULL
##
## alpha beta phi
## Geometric coefficients: 6672.885 -0.005863623 0.9716211checkresiduals(model3)## 2 3 4 5 6 7
## -253.202914 -30.610848 23.082621 -86.477237 -75.025302 12.803618
## 8 9 10 11 12 13
## 5.298155 -114.678300 18.261437 -170.880048 24.533185 -103.752494
## 14 15 16 17 18 19
## 8.852679 -32.170263 -83.952475 -27.466232 -2.098674 -56.865123
## 20 21 22 23 24 25
## -56.258419 41.535921 44.158599 92.935178 51.235104 771.792699
## 26 27 28 29 30 31
## -32.861153 176.884448 95.956039 -253.719314 38.523936 -95.592640
## 32 33 34 35 36 37
## 104.053203 35.245022 240.940477 115.938747 189.540536 117.568097
## 38 39 40 41 42 43
## 66.356304 175.905107 205.263342 213.917574 3.463459 -86.583649
## 44 45 46 47 48 49
## 91.002511 365.530625 242.621700 -141.692507 -135.968325 -689.639612
## 50 51 52 53 54 55
## -3.431458 -243.873540 262.447878 137.163954 -417.990992 -268.934588
## 56 57 58 59 60 61
## 161.681077 -584.660135 -677.835397 -364.478905 -80.335251 -247.606666
## 62 63 64 65 66 67
## -252.350187 161.705150 149.290705 12.444825 82.742609 255.096471
## 68 69 70 71 72 73
## 202.046722 229.415809 -110.296909 50.285608 152.562166 -293.406077
## 74 75 76 77 78 79
## 35.557299 228.407743 -64.382442 -392.363504 -153.655477 155.549369
## 80 81 82 83 84 85
## -87.231932 180.025045 87.330525 -62.445827 167.511796 7.078568
## 86 87 88 89 90 91
## 79.904245 11.079317 -23.496061 -108.066932 -129.399854 -159.845072
## 92 93 94 95 96 97
## -139.488905 -316.487992 259.891585 -197.000699 -99.959552 189.269180
## 98 99 100 101 102 103
## 45.284433 16.731182 31.851697 -349.789770 -26.704157 126.361650
## 104 105 106 107 108 109
## 25.995248 48.492013 112.049456 -32.844966 132.156466 226.632804
## 110 111 112 113 114 115
## 216.382610 -139.689291 182.996258 -254.784471 112.830231 -266.261379
## 116 117 118 119 120 121
## 338.187556 90.458999 203.539204 -100.085239 42.550965 -142.702300
## 122 123 124 125 126 127
## 210.564995 -9.092880 70.895949 9.292441 -85.832876 246.174124
## 128 129 130 131 132 133
## 12.765403 -317.254300 208.671523 -200.700159 89.282101 160.744259
## 134 135 136 137 138 139
## 348.671923 -23.746229 -75.816805 12.622825 -314.981263 227.827780
## 140 141 142 143 144 145
## -457.665890 -174.081084 214.773110 -81.539022 112.319064 -299.473074
## 146 147 148 149 150 151
## -127.601504 183.995301 149.606796 120.822879 -144.947561 323.454909
## 152 153 154 155 156 157
## -115.940495 -8.962756 -127.629175 95.901847 216.671093 -37.428029
## 158 159 160 161
## 92.511097 151.071498 55.297228 -174.052323
##
## Ljung-Box test
##
## data: Residuals
## Q* = 6.2327, df = 10, p-value = 0.7953
##
## Model df: 0. Total lags used: 10From the model 3 summary and Residual diagnostics we observe:
Independent variable X.t is not significant at 0.05 level.
Adjust R-squared is high at 0.9479 which means it explains 94.79% of the variability.
The p-value: 2.2e-16 < 0.05 therefore the model is significant.
The mean of the residuals is close to zero and there is no significant correlation in the residuals series.
ACF plot indicates no presence of a serial correlation in the residuals.
Histogram is slightly skewed but suggests normal distribution.
VIF.model3 = vif(model3$model)
VIF.model3## Y.1 X.t
## 1.493966 1.493966VIF.model3 > 10## Y.1 X.t
## FALSE FALSE- The value of VIF for model 3 is less than 10 therefore the effect of multicollinearity is low.
ARDLM
for(i in 1:5){
for(j in 1:5){
model4 = ardlDlm(x = as.vector(gold.ts)+as.vector(copper.ts), y = as.vector(asx.ts), p = i, q = j)
cat("p = ", i, "q = ", j, "AIC = ", AIC(model4$model), "BIC = ", BIC(model4$model), "\n")
}
}## p = 1 q = 1 AIC = 2150.525 BIC = 2165.901
## p = 1 q = 2 AIC = 2137.999 BIC = 2156.412
## p = 1 q = 3 AIC = 2123.806 BIC = 2145.244
## p = 1 q = 4 AIC = 2112.35 BIC = 2136.8
## p = 1 q = 5 AIC = 2101.551 BIC = 2129
## p = 2 q = 1 AIC = 2132.389 BIC = 2150.802
## p = 2 q = 2 AIC = 2134.313 BIC = 2155.796
## p = 2 q = 3 AIC = 2121.47 BIC = 2145.971
## p = 2 q = 4 AIC = 2109.799 BIC = 2137.305
## p = 2 q = 5 AIC = 2099.102 BIC = 2129.6
## p = 3 q = 1 AIC = 2119.28 BIC = 2140.718
## p = 3 q = 2 AIC = 2121.276 BIC = 2145.777
## p = 3 q = 3 AIC = 2121.607 BIC = 2149.17
## p = 3 q = 4 AIC = 2110.363 BIC = 2140.925
## p = 3 q = 5 AIC = 2099.587 BIC = 2133.135
## p = 4 q = 1 AIC = 2108.095 BIC = 2132.545
## p = 4 q = 2 AIC = 2110.063 BIC = 2137.569
## p = 4 q = 3 AIC = 2110.735 BIC = 2141.297
## p = 4 q = 4 AIC = 2112.075 BIC = 2145.693
## p = 4 q = 5 AIC = 2101.365 BIC = 2137.963
## p = 5 q = 1 AIC = 2097.433 BIC = 2124.881
## p = 5 q = 2 AIC = 2099.399 BIC = 2129.897
## p = 5 q = 3 AIC = 2100.132 BIC = 2133.68
## p = 5 q = 4 AIC = 2101.569 BIC = 2138.167
## p = 5 q = 5 AIC = 2103.364 BIC = 2143.013AIC and BIC are the lowest at p=5 and q=1
model4 = ardlDlm(x = as.vector(gold.ts)+as.vector(copper.ts), y = as.vector(asx.ts), p = 5, q = 1)
summary(model4)##
## Time series regression with "ts" data:
## Start = 6, End = 161
##
## Call:
## dynlm(formula = as.formula(model.text), data = data, start = 1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -657.92 -105.84 -0.14 126.82 754.76
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 224.844798 92.160670 2.440 0.0159 *
## X.t 0.069046 0.034666 1.992 0.0482 *
## X.1 -0.006477 0.054772 -0.118 0.9060
## X.2 -0.021511 0.055253 -0.389 0.6976
## X.3 -0.018254 0.055259 -0.330 0.7416
## X.4 -0.016303 0.054797 -0.298 0.7665
## X.5 -0.011201 0.034154 -0.328 0.7434
## Y.1 0.962947 0.021548 44.689 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 194.9 on 148 degrees of freedom
## Multiple R-squared: 0.9485, Adjusted R-squared: 0.946
## F-statistic: 389.1 on 7 and 148 DF, p-value: < 2.2e-16checkresiduals(model4$model)
##
## Breusch-Godfrey test for serial correlation of order up to 11
##
## data: Residuals
## LM test = 6.2206, df = 11, p-value = 0.8582From the Model 4 summary and residual diagnostics we observe:
No lags are significant at 5% level.
Adjust R-squared is high at 0.946 which means it may explains 94.6% of the variability.
The p-value: 2.2e-16 is < 0.05 therefore the model is significant.
The mean of the residuals is close to zero and there is no significant correlation in the residuals series
ACF plot indicates no presence of a serial correlation in the residuals.
Histogram is not normally distributed.
Breusch-Godfrey test p-value 0.8582 is > 0.05 therefore there is no significant autocorrelation remaining in the residuals.
VIF.model4 = vif(model4$model)
VIF.model4## X.t L(X.t, 1) L(X.t, 2) L(X.t, 3) L(X.t, 4) L(X.t, 5) L(y.t, 1)
## 24.431847 63.075733 66.339513 68.401097 69.191619 27.628706 1.366876VIF.model4 > 10## X.t L(X.t, 1) L(X.t, 2) L(X.t, 3) L(X.t, 4) L(X.t, 5) L(y.t, 1)
## TRUE TRUE TRUE TRUE TRUE TRUE FALSE- The value of VIF for model 4 is greater than 10 therefore the effect of multicollinearity is high.
Conclusion
Koyck DLM stands out as the most suited model for forecasting the ASX all Oridinaries (Ords) Price Index. This selection is justified by its high R-squared value of 0.9479, denoting its ability to explain 94.79% of the variability. Additionally, The residual series plot and acf of the residual suggest no sign of significant correlation in the series. The model also demonstrates low multicollinearity effect. Most importantly, the p-value is < 0.05 therefore the model is significant.