MATH1307 Forecasting Assignment 1
Ashwin Menon, StdID: S3808190
8/13/2021
Introduction
We use Time Series Analysis and Forecasting methods to draw statistical inferences about monthly averages of ASX all ordinaries (Ords) price index, Gold price, Crude Oil (Brent)_USD/bbl and Copper_USD/tonne starting from January 2004. The main aim of the analysis is to find a Distributed Lag Model (DLM) for the independent variable i.e. ASX Price Index. For the analysis, we used the asx_data from the RMIT University repository. The data set contains values of ASX price index, gold price in AUD, Crude oil per litre in USD and copper price per tonne in USD. We fit a time series plot for all the variables followed by analysing the stationarity and decomposition. In the last part, we fit four different models namely Distributed Lag Model (DLM), Polynomial DLM, Koyck DLM and Auto-regressive DLM for each variable. We compare each model using each models adjusted r-squared.
Loading necessary R-packages
##
## Attaching package: 'TSA'## The following objects are masked from 'package:stats':
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## acf, arima## The following object is masked from 'package:utils':
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## tar## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo## Loading required package: carData## Registered S3 methods overwritten by 'forecast':
## method from
## fitted.Arima TSA
## plot.Arima TSA## Loading required package: nardl## Loading required package: dynlm## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
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## as.Date, as.Date.numeric##
## Attaching package: 'dLagM'## The following object is masked from 'package:forecast':
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## forecast## Loading required package: x13binary## x12 is ready to use.## Use the package x12GUI for a Graphical User Interface.## By default the X13-ARIMA-SEATS binaries provided by the R package x13binary## are used but this can be changed with x12path(validpath)## ---------------## Suggestions and bug-reports can be submitted at: https://github.com/statistikat/x12/issues##
## Attaching package: 'readr'## The following object is masked from 'package:TSA':
##
## specLoading the dataset and converting it into a time series object
asx_data <- read_csv("ASX_data.csv")##
## -- Column specification --------------------------------------------------------
## cols(
## `ASX price` = col_double(),
## `Gold price` = col_number(),
## `Crude Oil (Brent)_USD/bbl` = col_double(),
## `Copper_USD/tonne` = col_number()
## )head(asx_data)## # A tibble: 6 x 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 1685class(asx_data)## [1] "spec_tbl_df" "tbl_df" "tbl" "data.frame"Creating individual time series object of each variable from the dataset
head(asx)## Jan Feb Mar Apr May Jun
## 2004 2935.4 2778.4 2848.6 2970.9 2979.8 2999.7head(gold)## Jan Feb Mar Apr May Jun
## 2004 611.9 603.3 565.7 538.6 549.4 535.9head(oil)## Jan Feb Mar Apr May Jun
## 2004 31.29 32.65 30.34 25.02 25.81 27.55head(copper)## Jan Feb Mar Apr May Jun
## 2004 1650 1682 1656 1588 1651 1685Checking the first five observations of the dataset after converting it into a time series object.
asx_price_index <- ts(asx_data, start = c(2004,1), frequency = 12)
class(asx_price_index)## [1] "mts" "ts" "matrix"head(asx_price_index)## ASX price Gold price Crude Oil (Brent)_USD/bbl Copper_USD/tonne
## Jan 2004 2935.4 611.9 31.29 1650
## Feb 2004 2778.4 603.3 32.65 1682
## Mar 2004 2848.6 565.7 30.34 1656
## Apr 2004 2970.9 538.6 25.02 1588
## May 2004 2979.8 549.4 25.81 1651
## Jun 2004 2999.7 535.9 27.55 1685A Time series plot of the dataset
asx.price_index_scaled = scale(asx_price_index)
plot(asx.price_index_scaled, plot.type="s",col = c("blue", "yellow", "red", "black"), main = "Fig 1. Visualizing monthly price of ASX, Gold, Crude oil, Copper")
legend("topleft",lty=1, text.width =1.5, col=c("blue", "yellow", "red", "black"), c("ASX", "Gold", "Oil", "Copper"))
Checking the existence of non-stationarity in the Dataset
The following time series plot of ASX price index starting from January 2004
plot(asx, ylab = "ASX Price Index", xlab = "Years", main = "Fig 2. Monthwise ASX price index starting from January 2004")
points(y = asx, x = time(asx), pch = as.vector(season(asx)))
Analysis & Interpretation
Trend- The ASX price index was increasing from 2004 to 2008. However, there is a sudden fall in the price index in 2008. But, again after 2008 we can see a upward trend.
Seasonality - Seasonality is present in the plot as we can observe that ASX price index increases in the months of February, March and April and decreases in the months of May and June.
Behaviour- Succeding points tell us the presence of an Auto-Regressive pattern in the time series plot.
Changing Variance - A change in variance is present.
Intervention Points- A major Intervention is observed during the year 2008 where the price index fell. It might be due to the great recession from 2007 to 2009.
ACF and PACF plots
Acf(asx, lag.max = 48, main = "Fig 3. ACF plot of ASX Price Index")
In the ACF Plot there are strong correlations between the elements of the time series. The lags seen are correlations of the observations of previous time stamps.We can conclude there is a presence of Seasonality in the time series data due to a wave like pattern in the ACF plot.
Pacf(asx, lag.max = 48 ,main = "Fig 4. PACF plot of ASX Price Index")
PACF plot has a significant spike only at lag 1. Hence, we say that all the higher-order autocorrelations are explained by the lag-1 autocorrelation
Augumented Dickey-Fuller Unit Root Test to check the existence of non-stationarity in the ASX price index.
summary(ur.df(asx, type="none", lags=1, selectlags="AIC"))##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression none
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -722.83 -103.12 36.63 138.79 809.17
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## z.lag.1 0.002107 0.003309 0.637 0.525
## z.diff.lag 0.085167 0.079885 1.066 0.288
##
## Residual standard error: 203.3 on 157 degrees of freedom
## Multiple R-squared: 0.01074, Adjusted R-squared: -0.001866
## F-statistic: 0.8519 on 2 and 157 DF, p-value: 0.4286
##
##
## Value of test-statistic is: 0.6366
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau1 -2.58 -1.95 -1.62From Augmented Dickey-Fuller Unit Root Test, the p_value = 0.4286 and value of test_statistic = 0.6366
As the p_value is greater that 5% level of significance, we fail to reject the null hypothesis. Hence, we assume that the data is non-stationary.
Further we test the assumption of existence of non-stationarity in the ASX price index using Phillips-Perron Unit Root Test
PP.test(asx,lshort = TRUE)##
## Phillips-Perron Unit Root Test
##
## data: asx
## Dickey-Fuller = -2.2284, Truncation lag parameter = 4, p-value = 0.4811PP.test(asx,lshort = FALSE)##
## Phillips-Perron Unit Root Test
##
## data: asx
## Dickey-Fuller = -2.3932, Truncation lag parameter = 13, p-value =
## 0.4123From the Phillips-Perron Unit Root Test, the p_value = 0.4811 when truncation lag parameter is 4 and p_value = 0.4123 when truncation lag parameter is 13
As the p_value is greater that 5% level of significance, we fail to reject the null hypothesis. Hence, we conclude that the data is non-stationary.
We apply a Box-Cox transformation to deal with non-stationarity in ASX price index
lambda_asx = BoxCox.lambda(asx, method = "loglik")
lambda_asx## [1] 1.05The value of lambda is 1.05
asx_boxcox <- BoxCox(asx, lambda = lambda_asx )
plot(asx_boxcox, type = "o", ylab = "ASX Price Index",
xlab = "Years", main = "Fig 5. Time series plot of ASX Price Index after BoxCox transformation")
Here, we can see no difference in the time series plot even after applying a box-cox transformation
This can be checked after applying adf and phillips-perron unit root test.
adf.test(asx_boxcox)##
## Augmented Dickey-Fuller Test
##
## data: asx_boxcox
## Dickey-Fuller = -2.6916, Lag order = 5, p-value = 0.2879
## alternative hypothesis: stationarypp.test(asx_boxcox)##
## Phillips-Perron Unit Root Test
##
## data: asx_boxcox
## Dickey-Fuller Z(alpha) = -8.7334, Truncation lag parameter = 4, p-value
## = 0.614
## alternative hypothesis: stationaryWe can see the p_value of both Augmented Dickey-Fuller and Phillips-perron Unit root test is greater than 5% level of significance. Hence we fail to reject the null hypothesis and conclude that the non-stationarity in the ASX price index still exists.
Applying first order ordinary differencing
asx_ordiff <- diff(asx)Time Series Plot after applying first ordinary differencing
plot(asx_ordiff, type = "o", ylab = "ASX Price Index",
xlab = "Years", main = "Fig 6. ASX Price Index with first order differencing")
Time series plot of the transformed data shows that the trend is reduced. But we can still intervention points. Also, the Auto-regressive pattern is still visible.
ACF and PACF plots of the transformed data
par(mfrow=c(1,2))
Acf(asx_ordiff, lag.max = 48, main = "Fig 7. ACF plot Transformed ASX price index")
Pacf(asx_ordiff, lag.max = 48, main = "Fig 8. PACF plot Transformed ASX price index")
par(mfrow=c(1,1))From the ACF and PACF plots we can see the decaying pattern has faded. There are no significant lags in the both the plots.
Augmented Dickey Fuller and Phillips-Perron Unit root test on the transformed data.
adf.test(asx_ordiff)## Warning in adf.test(asx_ordiff): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: asx_ordiff
## Dickey-Fuller = -4.5543, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationarypp.test(asx_ordiff)## Warning in pp.test(asx_ordiff): p-value smaller than printed p-value##
## Phillips-Perron Unit Root Test
##
## data: asx_ordiff
## Dickey-Fuller Z(alpha) = -162.52, Truncation lag parameter = 4, p-value
## = 0.01
## alternative hypothesis: stationaryFrom the above results, we can see the p_value for both Augmented Dickey Fuller and Phillips-Perron Unit root are less than 5% level of significance.
Hence, we reject the null hypothesis stating the transformed data is stationary.
We apply a X12 decomposition on the ASX price index time series data.
There are minute deviances from the trend component in the seasonally adjusted series. An increase in the ASX price index is due to effect of seasonality.
During the period of great recession i.e. from 2007 to 2009, we see a fall in the ASX price index which has also decreased the seasonality in the plot.
SI Ratios plot to check whether the seasonality flucuates around the mean
plotSeasFac(asx12, main = "Fig 10. SI Plot of ASX Price Index")
We observe seasonality this plot as the values fluctuate at the mean during the months of February, March and April followed by June, July and August.
We apply STL Decomposition on the transformed ASX price index time series data
asx_stl <- stl(asx, t.window = 15, s.window = "periodic",
robust = TRUE)
plot(asx_stl, main = "Fig 11. STL Plot of ASX Price Index")
In the above plot, we observe a smoothed plot of the original time series plot of ASX price index
The effect of seasonality is seen in the second plot.
Highest ASX price index values were recorded in the month of March while lowest ASX price index values were recorded in the months of October-November.
In the remainder part, there are some significant fluctuations and changing variance is present
We display the forecasts for ASX Price index from STL Method
asx_seasadj <- seasadj(asx_stl)
plot(naive(asx_seasadj), xlab ="New orders index",
main = "Fig 12. Naive forecasts of ASX Price Index")
Naive forecasts represent the mean level for ASX Price Index
The following time series plot of Gold price (AUD) starting from January 2004
plot(gold, ylab = "Gold Price", xlab = "Years", main = "Fig 13. Monthwise Gold price starting from January 2004")
points(y = gold, x = time(gold), pch = as.vector(season(gold)))
Analysis & Interpretation
Trend- There is an increasing trend in Gold price (AUD) since 2004.
Seasonality - We can observe seasonality in the plot the price of Gold is volatile. We observe prices fluctuating every year.
Behaviour- Succeding points tell us the presence of an Auto-Regressive pattern in the time series plot.
Changing Variance - A change in variance is present.
Intervention Points- No major Interventions are observed in the time series plot of Gold Price.
ACF and PACF plots
Acf(gold,lag.max = 48, main = "Fig 15. ACF plot of Gold Prices (AUD)")
In the ACF Plot there are strong correlations between the elements of the time series. The lags seen are correlations of the observations of previous time stamps.We can conclude there is a presence of Seasonality in the time series data due to a wave like pattern in the ACF plot.
Pacf(gold, lag.max = 48, main = "Fig 16. PACF plot of Gold Prices (AUD)")
PACF plot has a significant spike only at lag 1. Hence, we say that all the higher-order autocorrelations are explained by the lag-1 autocorrelation
Augumented Dickey-Fuller Unit Root Test to check the existence of non-stationarity in the Gold price (AUD)
summary(ur.df(gold, type="none", lags=1, selectlags="AIC"))##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression none
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -138.655 -26.586 -2.016 22.726 206.439
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## z.lag.1 0.003218 0.003322 0.969 0.334
## z.diff.lag 0.122451 0.079603 1.538 0.126
##
## Residual standard error: 52.57 on 157 degrees of freedom
## Multiple R-squared: 0.02351, Adjusted R-squared: 0.01107
## F-statistic: 1.89 on 2 and 157 DF, p-value: 0.1545
##
##
## Value of test-statistic is: 0.9687
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau1 -2.58 -1.95 -1.62From Augmented Dickey-Fuller Unit Root Test, the p_value = 0.1545 and value of test_statistic = 0.9687
As the p_value is greater that 5% level of significance, we fail to reject the null hypothesis. Hence, we assume that the data is non-stationary.
Further we test the assumption of existence of non-stationarity in the Gold price index using Phillips-Perron Unit Root Test
PP.test(gold,lshort = TRUE)##
## Phillips-Perron Unit Root Test
##
## data: gold
## Dickey-Fuller = -2.1044, Truncation lag parameter = 4, p-value = 0.5328PP.test(gold,lshort = FALSE)##
## Phillips-Perron Unit Root Test
##
## data: gold
## Dickey-Fuller = -1.8656, Truncation lag parameter = 13, p-value =
## 0.6324From the Phillips-Perron Unit Root Test, the p_value = 0.5328 when truncation lag parameter is 4 and p_value = 0.6324 when truncation lag parameter is 13
As the p_value is greater that 5% level of significance, we fail to reject the null hypothesis. Hence, we conclude that the data is non-stationary.
We apply a Box-Cox transformation to deal with non-stationarity in Gold price index
lambda_gold <- BoxCox.lambda(gold, method = "loglik")
lambda_gold## [1] 1.45The value of lambda is 1.05
gold_boxcox <- BoxCox(gold, lambda = lambda_gold)plot(gold_boxcox, type = "o", ylab = "Gold Price", xlab = "Years",
main = "Fig 17. Gold Prices (AUD) after Box-Cox transformation")
Here, we can see no difference in the time series plot even after applying a box-cox transformation
This can be checked after applying adf and phillips-perron unit root test.
adf.test(gold_boxcox)##
## Augmented Dickey-Fuller Test
##
## data: gold_boxcox
## Dickey-Fuller = -2.0813, Lag order = 5, p-value = 0.5424
## alternative hypothesis: stationarypp.test(gold_boxcox)##
## Phillips-Perron Unit Root Test
##
## data: gold_boxcox
## Dickey-Fuller Z(alpha) = -11.535, Truncation lag parameter = 4, p-value
## = 0.4534
## alternative hypothesis: stationaryWe can see the p_value of both Augmented Dickey-Fuller and Phillips-perron Unit root test is greater than 5% level of significance. Hence we fail to reject the null hypothesis and conclude that the non-stationarity in the Gold price (AUD) still exists.
Applying first order ordinary differencing
gold_ordiff <- diff(gold)Time Series Plot applying first ordinary differencing
plot(gold_ordiff, ylab = "GOld Price", xlab = "Years", type = "o",
main = "Fig 18. Gold Prices with first order differencing")
Time series plot of the transformed data shows that the trend is however reduced. But we can still intervention points. Also, the Auto-regressive pattern is still visible.
ACF and PACF plots of the transformed data
par(mfrow=c(1,2))
Acf(gold_ordiff,lag.max = 48, main = "Fig 19. ACF plot transformed Gold price (AUD)")
Pacf(gold_ordiff, lag.max = 48, main = "Fig 20. PACF plot transformed Gold price (AUD)")
par(mfrow=c(1,1))From the ACF and PACF plots we can see the decaying pattern has faded. There are no significant lags in the both the plots.
Augmented Dickey Fuller and Phillips-Perron Unit root test on the transformed data.
adf.test(gold_ordiff)## Warning in adf.test(gold_ordiff): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: gold_ordiff
## Dickey-Fuller = -5.8718, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationarypp.test(gold_ordiff)## Warning in pp.test(gold_ordiff): p-value smaller than printed p-value##
## Phillips-Perron Unit Root Test
##
## data: gold_ordiff
## Dickey-Fuller Z(alpha) = -127.96, Truncation lag parameter = 4, p-value
## = 0.01
## alternative hypothesis: stationaryFrom the above results, we can see the p_value for both Augmented Dickey Fuller and Phillips-Perron Unit root are less than 5% level of significance.
Hence, we reject the null hypothesis stating the transformed data is stationary.
We apply a X12 decomposition on the Gold price (AUD) time series data.
gold_x12 <- x12(gold)
plot(gold_x12, sa =TRUE, trend = TRUE, main = "Fig 21. X12 plot of Gold Price (AUD)")
There are minute deviances from the trend component in the seasonally adjusted series. An increase in the Gold price is due to effect of seasonality.
During the period of great recession i.e. from 2007 to 2009, we see a fall in the Gold price which has also decreased the seasonality in the plot.
SI Ratios plot to check whether the seasonality flucuates around the mean
plotSeasFac(gold_x12, main = "Fig 22. SI Plot of Gold Price (AUD)")
We observe seasonality this plot as the values fluctuate at the mean during the months of July,August and September followed by January, February and March.
We apply STL Decomposition on the transformed Gold price (AUD) time series data
gold_stl <- stl(gold, t.window = 15, s.window = "periodic",
robust = TRUE)
plot(gold_stl, main = "Fig 23. STL plot of Gold Price (AUD)")
In the above plot, we observe a smoothed plot of the original time series plot of Gold price
The effect of seasonality is seen in the second plot.
Highest Gold price values were recorded in the month of September while lowest Gold price values were recorded in the months of March
In the remainder part, there are some significant fluctuations and changing variance is present
We display the forecasts for Gold Price index from STL Method
gold_seasadj <- seasadj(gold_stl)
plot(naive(gold_seasadj), xlab = "New Orders Index",
main = "Fig 24. Naive forecasts of Gold Price (AUD)")
Naive forecasts represent the mean level for Gold Price Index
The following time series plot of Crude Oil starting from January 2004
plot(oil, ylab = "Crude Oil Price", xlab = "Years",
main = "Fig 25. Times Series plot of Crude Oil Prices per Litre (USD)")
points(y = oil, x = time(oil), pch = as.vector(season(oil)))
Analysis & Interpretation
Trend- We can a increasing trend in Crude Oil price since 2004. However, the crude oil suddenly falls in the year 2009. These trends are caused due the great recession.
Seasonality - A repeating cycle or pattern is be observed from the plot. The prices tend to increase during the February, March, April and prices fall during June, July. Hence, we can say there is a presence of seasonality.
Behaviour- Succeding points tell us the presence of an Auto-Regressive pattern in the time series plot.
Changing Variance - A change in variance is present.
Intervention Points- Two major Interventions are observed in the time series plot of Crude Oil Price. First is seen during the years 2007 to 2009 i.e. during the great recession period. The second intervention is seen in the year 2015.
ACF and PACF plots
Acf(oil,lag.max = 48, main = "Fig 26. ACF plot of Crude Oil Prices")
In the ACF Plot there are strong correlations between the elements of the time series. The lags seen are correlations of the observations of previous time stamps.We can conclude there is a presence of Seasonality in the time series data due to a wave like pattern in the ACF plot.
Pacf(oil,lag.max = 48, main = "Fig 27. ACF plot of Crude Oil Prices")
PACF plot has a significant spike only at lag 1. Hence, we say that all the higher-order autocorrelations are explained by the lag-1 autocorrelation
Augumented Dickey-Fuller Unit Root Test to check the existence of non-stationarity in the Crude Oil price
summary(ur.df(oil, type="none", lags=1, selectlags="AIC"))##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression none
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -19.905 -3.562 1.204 3.988 15.202
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## z.lag.1 -0.004013 0.005806 -0.691 0.49
## z.diff.lag 0.400123 0.073236 5.463 1.8e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.839 on 157 degrees of freedom
## Multiple R-squared: 0.1605, Adjusted R-squared: 0.1498
## F-statistic: 15.01 on 2 and 157 DF, p-value: 1.084e-06
##
##
## Value of test-statistic is: -0.6913
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau1 -2.58 -1.95 -1.62From Augmented Dickey-Fuller Unit Root Test, the p_value = 0.1498 and test_statistic = 0.6913
As the p_value is greater that 5% level of significance, we fail to reject the null hypothesis. Hence, we assume that the data is non-stationary.
Further we test the assumption of existence of non-stationarity in the Crude oil price using Phillips-Perron Unit Root Test
PP.test(oil,lshort = TRUE)##
## Phillips-Perron Unit Root Test
##
## data: oil
## Dickey-Fuller = -1.7567, Truncation lag parameter = 4, p-value = 0.6778PP.test(oil,lshort = FALSE)##
## Phillips-Perron Unit Root Test
##
## data: oil
## Dickey-Fuller = -1.5181, Truncation lag parameter = 13, p-value =
## 0.7773From the Phillips-Perron Unit Root Test, the p_value = 0.6778 when truncation lag parameter is 4 and p_value = 0.7773 when truncation lag parameter is 13
As the p_value is greater that 5% level of significance, we fail to reject the null hypothesis. Hence, we conclude that the data is non-stationary.
We apply a Box-Cox transformation to deal with non-stationarity in Gold price index
lambda_oil <- BoxCox.lambda(oil, method = "loglik")
lambda_oil## [1] 0.4oil_boxcox <- BoxCox(oil, lambda = lambda_oil)plot(oil_boxcox, type = "o", ylab = "Crude Oil Price", xlab = "Years",
main = "Fig 28. Crude Oil Prices per litre (USD) after Box-Cox transformation")
Here, we can see no difference in the time series plot even after applying a box-cox transformation
This can be checked after applying adf and phillips-perron unit root test.
adf.test(oil_boxcox)##
## Augmented Dickey-Fuller Test
##
## data: oil_boxcox
## Dickey-Fuller = -1.8707, Lag order = 5, p-value = 0.6303
## alternative hypothesis: stationarypp.test(oil_boxcox)##
## Phillips-Perron Unit Root Test
##
## data: oil_boxcox
## Dickey-Fuller Z(alpha) = -6.0377, Truncation lag parameter = 4, p-value
## = 0.7685
## alternative hypothesis: stationaryWe can see the p_value of both Augmented Dickey-Fuller and Phillips-perron Unit root test is greater than 5% level of significance. Hence we fail to reject the null hypothesis and conclude that the non-stationarity in the Crude oil price still exists.
Applying first order ordinary differencing
oil_ordiff <- diff(oil)Time Series Plot applying first ordinary differencing
plot(oil_ordiff, ylab = "Crude Oil Price", xlab = "Years", type = "o",
main = "Fig 29. Crude Oil Prices per litre (USD) with first order differencing")
Time series plot of the transformed data shows that the trend is reduced. However, we can still intervention points. Also, the Auto-regressive pattern is still visible.
ACF and PACF plots of the transformed data
par(mfrow=c(1,2))
Acf(oil_ordiff, lag.max = 48,main = "Fig 30. ACF plot transformed Crude oil price per litre (USD)")
Pacf(oil_ordiff,lag.max = 48, main = "Fig 31. PACF plot transformed Crude oil price per litre (USD)")
par(mfrow=c(1,1))From the ACF and PACF plots we can see the decaying pattern has faded but some part of it is still visible. We can there are no significant lags at lag-1 in the both the plots.
Augmented Dickey Fuller and Phillips-Perron Unit root test on the transformed data.
adf.test(oil_ordiff)## Warning in adf.test(oil_ordiff): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: oil_ordiff
## Dickey-Fuller = -5.4261, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationarypp.test(oil_ordiff)## Warning in pp.test(oil_ordiff): p-value smaller than printed p-value##
## Phillips-Perron Unit Root Test
##
## data: oil_ordiff
## Dickey-Fuller Z(alpha) = -96.402, Truncation lag parameter = 4, p-value
## = 0.01
## alternative hypothesis: stationaryFrom the above results, we can see the p_value for both Augmented Dickey Fuller and Phillips-Perron Unit root are less than 5% level of significance.
Hence, we reject the null hypothesis stating the transformed data is stationary.
We apply a X12 decomposition on the Gold price (AUD) time series data.
oil_x12 <- x12(oil)
plot(oil_x12, sa = TRUE, trend = TRUE, main = "Fig 32. X12 plot of Crude Oil Price per litre (USD)")
There are minute deviances from the trend component in the seasonally adjusted series. An increase in the Crude Oil price is due to effect of seasonality.
During the period of great recession i.e. from 2007 to 2009, we see a fall in the Crude oil price which has also decreased the seasonality in the plot.
SI Ratios plot to check whether the seasonality flucuates around the mean
plotSeasFac(oil_x12, main = "Fig 33. SI Plot of Crude Oil Price per litre (USD)")
We observe seasonality this plot as the values fluctuate at the mean during the months of February and March followed by June and July.
We apply STL Decomposition on the transformed Gold price (AUD) time series data
oil_stl <- stl(oil, t.window = 15, s.window = "periodic",
robust = TRUE)
plot(oil_stl, main = "Fig 34. STL Plot of Crude Oil Price per litre (USD)")
In the above plot, we observe a smoothed plot of the original time series plot of Crude oil price
The effect of seasonality is seen in the second plot.
Highest Crude oil price values were recorded in the month of March while lowest Crude oil price values were recorded in the months of July
In the remainder part, there are some significant fluctuations and changing variance is present
We display the forecasts for Crude oil Price from STL Method
oil_seasadj <- seasadj(oil_stl)
plot(naive(oil_seasadj), xlab = "New Orders Index",
main = "Fig 35. Naive forecasts of Crude Oil Price per litre (USD)")
Naive forecasts represent the mean level for Crude oil Price
The following time series plot of Crude Oil starting from January 2004
plot(copper, ylab = "Copper Price", xlab = "Years",
main = "Fig 36. Times Series plot of Copper Price per tonne (USD)")
points(y = copper, x = time(copper), pch = as.vector(season(copper)))
Analysis & Interpretation
Trend- We can a increasing trend in Copper price since 2004. However, the copper price suddenly falls in the year 2009. These trends are caused due the great recession.
Seasonality - A repeating cycle or pattern is be observed in the plot. The prices tend to increase during the December, January, February and prices fall during August, September and October. Hence, we can say there is a presence of seasonality.
Behaviour- Succeding points tell us the presence of an Auto-Regressive pattern in the time series plot.
Changing Variance - A change in variance is present.
Intervention Points- A major Intervention is observed in the time series plot of Copper Price. It is seen during the years 2007 to 2009 i.e. during the great recession period.
ACF and PACF plots
Acf(copper,lag.max = 48, main = "Fig 37. ACF plot of Copper Prices")
In the ACF Plot there are strong correlations between the elements of the time series. The lags seen are correlations of the observations of previous time stamps.We can conclude there is a presence of Seasonality in the time series data due to a wave like pattern in the ACF plot.
Pacf(copper,lag.max = 48, main = "Fig 38. ACF plot of Copper Prices")
PACF plot has a significant spike at lag 1. Hence, we say that all the higher-order autocorrelations are explained by the lag-1 autocorrelation
Augumented Dickey-Fuller Unit Root Test to check the existence of non-stationarity in the Copper price
summary(ur.df(copper, type="none", lags=1, selectlags="AIC"))##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression none
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1861.58 -164.87 32.72 248.08 2829.69
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## z.lag.1 -0.002256 0.005772 -0.391 0.696
## z.diff.lag 0.307564 0.076139 4.040 8.36e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 463.1 on 157 degrees of freedom
## Multiple R-squared: 0.09422, Adjusted R-squared: 0.08268
## F-statistic: 8.166 on 2 and 157 DF, p-value: 0.0004229
##
##
## Value of test-statistic is: -0.3909
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau1 -2.58 -1.95 -1.62adf.test(copper)##
## Augmented Dickey-Fuller Test
##
## data: copper
## Dickey-Fuller = -2.2502, Lag order = 5, p-value = 0.472
## alternative hypothesis: stationaryFrom Augmented Dickey-Fuller Unit Root Test, the p_value = 0.472 and test_statistic = 2.2502
The p_value is greater that 5% level of significance, hence, we fail reject the null hypothesis. We assume that the data is non-stationary.
Further we test the assumption of existence of non-stationarity in the Crude oil price using Phillips-Perron Unit Root Test
PP.test(copper,lshort = TRUE)##
## Phillips-Perron Unit Root Test
##
## data: copper
## Dickey-Fuller = -2.0718, Truncation lag parameter = 4, p-value = 0.5464PP.test(copper,lshort = FALSE)##
## Phillips-Perron Unit Root Test
##
## data: copper
## Dickey-Fuller = -1.9005, Truncation lag parameter = 13, p-value =
## 0.6178From the Phillips-Perron Unit Root Test, the p_value = 0.5464 when truncation lag parameter is 4 and p_value = 0.6178 when truncation lag parameter is 13
As the p_value is greater that 5% level of significance, we fail to reject the null hypothesis. Hence, we conclude that the data is non-stationary.
We apply a Box-Cox transformation to deal with non-stationarity in Copper price index
lambda_copper <- BoxCox.lambda(copper, method = "loglik")
lambda_copper## [1] 1.1The value of lambda is 1.1
copper_boxcox<- BoxCox(copper, lambda = lambda_copper)plot(copper_boxcox, type = "o", ylab = "Copper Price", xlab = "Years",
main = "Fig 40. Copper Prices with Box-Cox transformation")
Here, we can see no difference in the time series plot even after applying a box-cox transformation
This can be checked after applying adf and phillips-perron unit root test.
adf.test(copper_boxcox)##
## Augmented Dickey-Fuller Test
##
## data: copper_boxcox
## Dickey-Fuller = -2.2362, Lag order = 5, p-value = 0.4778
## alternative hypothesis: stationarypp.test(copper_boxcox)##
## Phillips-Perron Unit Root Test
##
## data: copper_boxcox
## Dickey-Fuller Z(alpha) = -7.995, Truncation lag parameter = 4, p-value
## = 0.6563
## alternative hypothesis: stationaryWe can see the p_values of both Augmented Dickey-Fuller and Phillips-perron Unit root test is greater than 5% level of significance. Hence we fail to reject the null hypothesis and conclude that the non-stationarity in the Copper price still exists.
Applying first order ordinary differencing
copper_ordiff <- diff(copper)Time Series Plot applying first ordinary differencing
plot(copper_ordiff, ylab = "Copper Price", xlab = "Years", type = "o",
main = "Fig 41. Copper Prices with first order differencing")
Time series plot of the transformed data shows that the trend is reduced. However, we can still intervention points. Also, the Auto-regressive pattern is still visible.
ACF and PACF plots of the transformed data
par(mfrow=c(1,2))
Acf(copper_ordiff,lag.max = 48, main = "Fig 42. ACF plot transformed Copper price index")
Pacf(copper_ordiff,lag.max = 48, main = "Fig 43. PACF plot transformed Copper price index")
par(mfrow=c(1,1))From the ACF and PACF plots we can see the decaying pattern has faded but some part of it is still visible. We can there are no significant lags at lag-1 in the both the plots.
Augmented Dickey Fuller and Phillips-Perron Unit root test on the transformed data.
adf.test(copper_ordiff)## Warning in adf.test(copper_ordiff): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: copper_ordiff
## Dickey-Fuller = -5.478, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationarypp.test(copper_ordiff)## Warning in pp.test(copper_ordiff): p-value smaller than printed p-value##
## Phillips-Perron Unit Root Test
##
## data: copper_ordiff
## Dickey-Fuller Z(alpha) = -106.69, Truncation lag parameter = 4, p-value
## = 0.01
## alternative hypothesis: stationaryFrom the above results, we can see the p_value for both Augmented Dickey Fuller and Phillips-Perron Unit root are less than 5% level of significance.
Hence, we reject the null hypothesis stating the transformed data is stationary.
We apply a X12 decomposition on the Copper price time series data.
copper_x12 <- x12(copper)
plot(copper_x12, sa = TRUE, trend = TRUE, main = "Fig 44. X12 Plot of Copper price per tonne (USD)")
There are minute deviances from the trend component in the seasonally adjusted series. An increase in the Copper price is due to effect of seasonality.
During the period of great recession i.e. from 2007 to 2009, we see a fall in the Copper price which has also decreased the seasonality in the plot.
SI Ratios plot to check whether the seasonality flucuates around the mean
plotSeasFac(copper_x12, main = "Fig 45. SI Plot of Copper price per tonne (USD)")
We observe seasonality this plot as the values fluctuate at the mean during the months of January and February followed by September and October.
We apply STL Decomposition on the transformed Copper price time series data
copper_stl <- stl(copper, t.window = 15, s.window = "periodic",
robust = TRUE)
plot(copper_stl, main = "Fig 46. STL Plot of Copper price per tonne (USD)")
In the above plot, we observe a smoothed plot of the original time series plot of Crude oil price
The effect of seasonality is seen in the second plot.
Highest Copper price values were recorded in the month of February while lowest Copper price values were recorded in the months of October
In the remainder part, there are some significant fluctuations and changing variance is present
We display the forecasts for Copper Price from STL Method
copper_seasadj <- seasadj(copper_stl)
plot(naive(copper_seasadj), xlab = "New Orders Index",
main = "Fig 47. Naive forecasts of Copper Prices per tonne (USD)")
Naive forecasts represent the mean level for Copper Price
A Time Series plot of the Data Set
plot(asx_price_index, xlab = "Years", ylab = "Copper Crude_Oil Gold ASX",
main = "Fig 48. Time series plot for ASXPrice Index, Gold (AUD), Crude Oil (USD/litre) and Copper Price (USD/tonne")
A Pair plot (Scatterplot) to check correlation between the dependent and independent variable
pairs(asx_price_index[,1:4], pch = 1, lower.panel = NULL)
To further check this, we use cor() function.
cor(asx_price_index)## ASX price Gold price Crude Oil (Brent)_USD/bbl
## ASX price 1.0000000 0.3431908 0.3290338
## Gold price 0.3431908 1.0000000 0.4366382
## Crude Oil (Brent)_USD/bbl 0.3290338 0.4366382 1.0000000
## Copper_USD/tonne 0.5617864 0.5364213 0.8664296
## Copper_USD/tonne
## ASX price 0.5617864
## Gold price 0.5364213
## Crude Oil (Brent)_USD/bbl 0.8664296
## Copper_USD/tonne 1.0000000A strong positive correlation is observed between Copper_USD/tonne and ASX price
From this, we can explain the dependent variable ASX price index more precisely when we use the information from the independent variable Copper_USD/tonne
Firstly, we fit a distributed lag model for ASX price and Copper.
asx_copper = data.frame(asx_data$`ASX price`,asx_data$`Copper_USD/tonne`)
asx_copper.ts = ts(asx_copper, start = 2004, frequency = 12)
head(asx_copper.ts)## asx_data..ASX.price. asx_data..Copper_USD.tonne.
## Jan 2004 2935.4 1650
## Feb 2004 2778.4 1682
## Mar 2004 2848.6 1656
## Apr 2004 2970.9 1588
## May 2004 2979.8 1651
## Jun 2004 2999.7 1685asx_copper_scaled = scale(asx_copper.ts)
plot(asx_copper_scaled, plot.type = "s", col = c("blue","red"), main = "Fig 49. Time Series Plot of Copper price per tonne (USD) vs ASX Price Index")
legend("bottomright", colnames(asx_copper_scaled), col=1:ncol(asx_copper), lty=2, cex=.75)
Distrubuted Lag Model of ASX price index and copper_usd/tonne with lag 1
model1c <- dlm(x = as.vector(copper), y = as.vector(asx),
q = 1)
summary(model1c)##
## Call:
## lm(formula = model.formula, data = design)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1315.8 -679.5 -117.9 646.4 1506.1
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.437e+03 1.781e+02 19.295 <2e-16 ***
## x.t 1.847e-01 1.238e-01 1.492 0.138
## x.1 4.624e-02 1.222e-01 0.378 0.706
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 743.1 on 157 degrees of freedom
## Multiple R-squared: 0.3028, Adjusted R-squared: 0.2939
## F-statistic: 34.09 on 2 and 157 DF, p-value: 5.06e-13
##
## AIC and BIC values for the model:
## AIC BIC
## 1 2574.488 2586.789From the above model, we observe the following
Residual standard error = 743.1 on 157 degrees of freedom
The model is significant as the p_value is less 5% level of significance
F-statistic of the model = 34.09
The multiple and adjusted R_sqrd are 30.28 and 29.39 respectively. However the adjusted R_sqrd is very small
Also, no variables except the intercept in the model are significant as they are greater than 5% level of significance
Residual check and Serial Correlation Check using Breusch-Godfrey
checkresiduals(model1c$model)
##
## Breusch-Godfrey test for serial correlation of order up to 10
##
## data: Residuals
## LM test = 147.66, df = 10, p-value < 2.2e-16From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, p_value of breusch-godfrey test is less that 5% level of significance. This concludes serial correlation in the residuals is highly significant.
Multi-collinearity Check
vif_model1c <- vif(model1c$model)
vif_model1c## x.t x.1
## 19.66222 19.66222vif_model1c > 10## x.t x.1
## TRUE TRUEAs the Variation inflation factor is greater than 10, we conclude the effect of multi-collinearity is high
Specification of lag length:
model1c_AIC_lag_length <- finiteDLMauto(x = as.vector(copper), y = as.vector(asx),
q.min = 1, q.max = 12, model.type = "dlm",
error.type = "AIC", trace = TRUE)
model1c_AIC_lag_length## q - k MASE AIC BIC GMRAE MBRAE R.Adj.Sq Ljung-Box
## 12 12 3.88750 2405.162 2450.221 4.98966 0.75956 0.08924 0
## 11 11 3.93602 2420.623 2462.771 5.00068 1.10787 0.10850 0
## 10 10 3.97380 2436.164 2475.389 5.07172 0.93641 0.12924 0
## 9 9 4.00524 2451.016 2487.302 5.07902 0.99736 0.14957 0
## 8 8 4.03682 2466.511 2499.846 5.19900 0.04304 0.16793 0
## 7 7 4.07595 2481.988 2512.357 5.19017 0.78489 0.18422 0
## 6 6 4.10610 2497.775 2525.166 5.29119 0.78994 0.20021 0
## 5 5 4.12565 2513.265 2537.664 5.36259 0.90478 0.21921 0
## 4 4 4.16156 2528.895 2550.289 5.38475 0.88789 0.23649 0
## 3 3 4.19532 2544.155 2562.531 5.23016 0.67751 0.25434 0
## 2 2 4.21688 2559.356 2574.700 5.34802 0.67093 0.27395 0
## 1 1 4.24262 2574.488 2586.789 5.38993 -0.49453 0.29391 0model1c_BIC_lag_length <-
finiteDLMauto(x =as.vector(copper), y = as.vector(asx),
q.min = 1, q.max = 12, error.type = "BIC",
model.type = "dlm", trace = TRUE)
model1c_BIC_lag_length## q - k MASE AIC BIC GMRAE MBRAE R.Adj.Sq Ljung-Box
## 12 12 3.88750 2405.162 2450.221 4.98966 0.75956 0.08924 0
## 11 11 3.93602 2420.623 2462.771 5.00068 1.10787 0.10850 0
## 10 10 3.97380 2436.164 2475.389 5.07172 0.93641 0.12924 0
## 9 9 4.00524 2451.016 2487.302 5.07902 0.99736 0.14957 0
## 8 8 4.03682 2466.511 2499.846 5.19900 0.04304 0.16793 0
## 7 7 4.07595 2481.988 2512.357 5.19017 0.78489 0.18422 0
## 6 6 4.10610 2497.775 2525.166 5.29119 0.78994 0.20021 0
## 5 5 4.12565 2513.265 2537.664 5.36259 0.90478 0.21921 0
## 4 4 4.16156 2528.895 2550.289 5.38475 0.88789 0.23649 0
## 3 3 4.19532 2544.155 2562.531 5.23016 0.67751 0.25434 0
## 2 2 4.21688 2559.356 2574.700 5.34802 0.67093 0.27395 0
## 1 1 4.24262 2574.488 2586.789 5.38993 -0.49453 0.29391 0We see from the AIC and BIC criteria that lag 12 has the lowest AIC and BIC values. Hence, we specify the lag length as 12.
model1.1c <- dlm(x = as.vector(copper), y = as.vector(asx), q = 12)
summary(model1.1c)##
## Call:
## lm(formula = model.formula, data = design)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1196.59 -618.58 58.35 582.94 1441.94
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.117e+03 2.276e+02 18.087 <2e-16 ***
## x.t 1.458e-01 1.369e-01 1.065 0.289
## x.1 2.374e-02 2.212e-01 0.107 0.915
## x.2 3.433e-02 2.239e-01 0.153 0.878
## x.3 2.489e-02 2.234e-01 0.111 0.911
## x.4 2.331e-02 2.225e-01 0.105 0.917
## x.5 -4.938e-02 2.209e-01 -0.224 0.823
## x.6 3.183e-02 2.207e-01 0.144 0.886
## x.7 -1.153e-02 2.218e-01 -0.052 0.959
## x.8 -2.387e-03 2.244e-01 -0.011 0.992
## x.9 -6.943e-02 2.252e-01 -0.308 0.758
## x.10 -2.573e-02 2.258e-01 -0.114 0.909
## x.11 -2.364e-02 2.232e-01 -0.106 0.916
## x.12 2.764e-02 1.357e-01 0.204 0.839
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 735.6 on 135 degrees of freedom
## Multiple R-squared: 0.1692, Adjusted R-squared: 0.08924
## F-statistic: 2.115 on 13 and 135 DF, p-value: 0.01687
##
## AIC and BIC values for the model:
## AIC BIC
## 1 2405.162 2450.221From the above model, we observe the following
Residual standard error = 735.6 on 135 degrees of freedom
The model is significant as the p_value is less 5% level of significance
F-statistic of the model = 2.115
The multiple and adjusted R_sqrd are 16.92 and 8.924 respectively. However the adjusted R_sqrd is very small
Also, no variables except the intercept in the model are significant as they are greater than 5% level of significance
Residual check and Serial Correlation Check using Breusch-Godfrey
checkresiduals(model1.1c$model)
##
## Breusch-Godfrey test for serial correlation of order up to 17
##
## data: Residuals
## LM test = 137.27, df = 17, p-value < 2.2e-16From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, p_value of breusch-godfrey test is less that 5% level of significance. This concludes serial correlation in the residuals is highly significant.
Multi-collinearity Check
vif_model1.1c <- vif(model1.1c$model)
vif_model1.1c## x.t x.1 x.2 x.3 x.4 x.5 x.6 x.7
## 17.21011 46.38555 49.15297 50.67591 52.07731 53.10852 54.77615 57.11949
## x.8 x.9 x.10 x.11 x.12
## 60.14983 62.29806 64.29926 64.49842 24.41346vif_model1.1c > 10## x.t x.1 x.2 x.3 x.4 x.5 x.6 x.7 x.8 x.9 x.10 x.11 x.12
## TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUEAs the Variation inflation factor is greater than 10, we conclude the effect of multi-collinearity is high
Polynomial Distributed Lag Model
We can reduce the effect of multi-collinearity using the polynomial distributed lag model
model2c <- polyDlm(x = as.vector(copper), y = as.vector(asx),
q = 1, k = 1, show.beta = FALSE)
summary(model2c)##
## Call:
## "Y ~ (Intercept) + X.t"
##
## Residuals:
## Min 1Q Median 3Q Max
## -1315.8 -679.5 -117.9 646.4 1506.1
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3436.7102 178.1110 19.295 <2e-16 ***
## z.t0 0.1847 0.1238 1.492 0.138
## z.t1 -0.1385 0.2445 -0.566 0.572
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 743.1 on 157 degrees of freedom
## Multiple R-squared: 0.3028, Adjusted R-squared: 0.2939
## F-statistic: 34.09 on 2 and 157 DF, p-value: 5.06e-13From the above model, we observe the following
Residual standard error = 743.1 on 157 degrees of freedom
The model is significant as the p_value is less 5% level of significance
F-statistic of the model = 34.09
The multiple and adjusted R_sqrd are 30.28 and 29.39 respectively. However the adjusted R_sqrd is very small
Also, no variables except the intercept in the model are significant as they are greater than 5% level of significance
A polynomial pattern (q = 4, k = 4) is imposed on the lag distribution i.e. equivalent to represent beta parameters with Kth order polynomial model time.
model2.2c <- polyDlm(x = as.vector(copper), y = as.vector(asx),
q = 4, k = 4, show.beta = TRUE)## Estimates and t-tests for beta coefficients:
## Estimate Std. Error t value P(>|t|)
## beta.0 0.1670 0.131 1.270 0.205
## beta.1 0.0332 0.211 0.157 0.875
## beta.2 0.0405 0.213 0.190 0.849
## beta.3 0.0645 0.211 0.306 0.760
## beta.4 -0.0980 0.129 -0.758 0.449In the above polynomial distributed lag model, we observe that all the variables are not significant to 5% level of significance.
Residual check and Serial Correlation Check using Breusch-Godfrey
checkresiduals(model2.2c$model)
##
## Breusch-Godfrey test for serial correlation of order up to 10
##
## data: Residuals
## LM test = 144.76, df = 10, p-value < 2.2e-16From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, p_value of breusch-godfrey test is less that 5% level of significance. This concludes serial correlation in the residuals is highly significant.
Multi-collinearity Check
vif_model2.2c <- vif(model2.2c$model)
vif_model2.2c## z.t0 z.t1 z.t2 z.t3 z.t4
## 506.9064 319571.6069 4844004.8499 8794411.8246 1726358.6343vif_model2.2c > 10## z.t0 z.t1 z.t2 z.t3 z.t4
## TRUE TRUE TRUE TRUE TRUEAs the Variation inflation factor is greater than 10, we conclude the effect of multi-collinearity is high
Koyck Distributed Lag Model
model3c <- koyckDlm(x = as.vector(copper), y = as.vector(asx))
summary(model3c, diagnostics = TRUE)##
## 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:
## df1 df2 statistic p-value
## Weak instruments 1 157 1966.86799 1.043205e-90
## Wu-Hausman 1 156 10.97528 1.147725e-03
##
## alpha beta phi
## Geometric coefficients: 6672.885 -0.005863623 0.9716211From the above model, we observe the following
Residual standard error = 201.9 on 157 degrees of freedom
The model is significant as the p_value is less 5% level of significance
The multiple and adjusted R_sqrd are 94.85 and 94.79 respectively. The adjusted R_sqrd are comparitively greater than the Distributed Lag & Polynomial Distributed Lag model
Also, variable Y.1 in the model is significant to 5% level of significance. The beta and phi values are -0.0058 and 0.971 respectively
The weak instruments p_value suggests that the model is significant at first stage of least squares fitting
Wu-Hausman test is used to compare the consistency of instrumental variable estimates to ordinary least square estimates. The presence of endogeniety tells us that instrumental variable estimate is consistent and ordinary least square estimate is inconsistent.
The Wu-Hausman test is significant as the p_value is less than 5% level of significance. Hence, we reject the null hypothesis that correlation between explanatory vairable and error term is zero or in other words endogeniety is not present.
Therefore, we conclude that there is significant correlation between explanatory variable and error term is present at 5% level of significance.
Residual Check
checkresiduals(model3c$model)
From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Multi-collinearity Check
vif_model3c <- vif(model3c$model)
vif_model3c## Y.1 X.t
## 1.493966 1.493966vif_model3c > 10## Y.1 X.t
## FALSE FALSEThe variation inflation factor is less than 5 when value of q = 2, we conclude the effect of multi-collinearity is not present.
Auto Regressive Distributed Lag Model
model4c <- ardlDlm(x = as.vector(copper), y = as.vector(asx),
p = 1, q = 1)
summary(model4c)##
## Time series regression with "ts" data:
## Start = 2, End = 161
##
## Call:
## dynlm(formula = as.formula(model.text), data = data, start = 1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -734.34 -101.82 16.56 123.05 774.55
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 173.61066 84.94572 2.044 0.04266 *
## X.t 0.10629 0.03258 3.263 0.00136 **
## X.1 -0.10695 0.03228 -3.313 0.00115 **
## Y.1 0.96784 0.02103 46.027 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 195.2 on 156 degrees of freedom
## Multiple R-squared: 0.9522, Adjusted R-squared: 0.9513
## F-statistic: 1035 on 3 and 156 DF, p-value: < 2.2e-16From the above model, we observe the following
Residual standard error = 195.2 on 156 degrees of freedom
The model is significant as the p_value is less 5% level of significance
F-statistic of the model = 1035 on 3 and 156 Degrees of freedom
The multiple and adjusted R_sqrd are 95.22 and 95.13 respectively. The adjusted R_sqrd of this model is comparitively higher than DLM, Polynomial DLM and Koyck models.
Also, all the variables in the model are significant as they are smaller than 5% level of significance
Residuals Check
checkresiduals(model4c$model, lag = 2)
##
## Breusch-Godfrey test for serial correlation of order up to 2
##
## data: Residuals
## LM test = 2.1004, df = 2, p-value = 0.3499The breusch-godfrey test has a p_value greater than 5% level of significance. Hence, it states we reject the null hypothesis that variance is not changing in the residuals.
From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, we can infer that the standardized residuals in the histogram are distributed around zero mean levels
The ACF plot of the residuals do not suggest the existence of Auto-correlation within error terms
Multi-collinearity Check
vif_model4c <- vif(model4c$model)
vif_model4c## X.t L(X.t, 1) L(y.t, 1)
## 19.716144 19.873449 1.473109vif_model4c > 10## X.t L(X.t, 1) L(y.t, 1)
## TRUE TRUE FALSEThe variation inflation factor is less than 5 for coefficient Y.t when value of q = 1.
Determining the values of p and q using AIC and BIC values
for(i in 1:5){
for(j in 1:5){
model4.1c <- ardlDlm(x = as.vector(copper),
y = as.vector(asx), p = i, q = j)
cat("p:",i,"q:",j,"AIC:",AIC(model4.1c$model), "BIC:",
BIC(model4.1c$model),"\n")
}
}## p: 1 q: 1 AIC: 2147.741 BIC: 2163.116
## p: 1 q: 2 AIC: 2135.4 BIC: 2153.813
## p: 1 q: 3 AIC: 2121.12 BIC: 2142.558
## p: 1 q: 4 AIC: 2109.759 BIC: 2134.209
## p: 1 q: 5 AIC: 2099.056 BIC: 2126.505
## p: 2 q: 1 AIC: 2130.043 BIC: 2148.456
## p: 2 q: 2 AIC: 2132.038 BIC: 2153.52
## p: 2 q: 3 AIC: 2119.241 BIC: 2143.741
## p: 2 q: 4 AIC: 2107.649 BIC: 2135.155
## p: 2 q: 5 AIC: 2097.021 BIC: 2127.52
## p: 3 q: 1 AIC: 2117.307 BIC: 2138.745
## p: 3 q: 2 AIC: 2119.247 BIC: 2143.748
## p: 3 q: 3 AIC: 2119.696 BIC: 2147.259
## p: 3 q: 4 AIC: 2108.537 BIC: 2139.1
## p: 3 q: 5 AIC: 2097.832 BIC: 2131.38
## p: 4 q: 1 AIC: 2105.916 BIC: 2130.366
## p: 4 q: 2 AIC: 2107.774 BIC: 2135.28
## p: 4 q: 3 AIC: 2108.608 BIC: 2139.17
## p: 4 q: 4 AIC: 2110.085 BIC: 2143.704
## p: 4 q: 5 AIC: 2099.454 BIC: 2136.052
## p: 5 q: 1 AIC: 2095.118 BIC: 2122.566
## p: 5 q: 2 AIC: 2096.96 BIC: 2127.459
## p: 5 q: 3 AIC: 2097.887 BIC: 2131.436
## p: 5 q: 4 AIC: 2099.497 BIC: 2136.095
## p: 5 q: 5 AIC: 2101.419 BIC: 2141.067We observe that p = 5 and q = 1 give the smallest AIC and BIC values.
best_model_copper <- ardlDlm(x = as.vector(copper), y = as.vector(asx),
p = 5, q = 1)
summary(best_model_copper)##
## 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
## -663.01 -104.28 -6.52 134.04 748.83
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 223.07851 91.31917 2.443 0.0157 *
## X.t 0.08180 0.03445 2.374 0.0189 *
## X.1 -0.02382 0.05497 -0.433 0.6654
## X.2 -0.02426 0.05554 -0.437 0.6629
## X.3 -0.01059 0.05554 -0.191 0.8490
## X.4 -0.01423 0.05498 -0.259 0.7962
## X.5 -0.01564 0.03396 -0.461 0.6457
## Y.1 0.96496 0.02137 45.160 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 193.4 on 148 degrees of freedom
## Multiple R-squared: 0.9492, Adjusted R-squared: 0.9468
## F-statistic: 395.2 on 7 and 148 DF, p-value: < 2.2e-16From the above model, we observe the following
Residual standard error = 193.4 on 148 degrees of freedom
The model is significant as the p_value is less 5% level of significance
F-statistic of the model = 395.2 on 7 and 148 Degrees of freedom
The multiple and adjusted R_sqrd are 94.92 and 94.68 respectively. The adjusted R_sqrd of this model is comparitively higher than DLM, Polynomial DLM and Koyck models.
Variables X.t and Y.1 in the model are significant as they are smaller than 5% level of significance
Residuals Check
checkresiduals(best_model_copper$model)
##
## Breusch-Godfrey test for serial correlation of order up to 11
##
## data: Residuals
## LM test = 5.0114, df = 11, p-value = 0.9306The breusch-godfrey test has a p_value greater than 5% level of significance. Hence, it states we reject the null hypothesis that variance is not changing in the residuals.
From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, we can infer that the standardized residuals in the histogram are distributed around zero mean levels
The ACF plot of the residuals do not suggest the existence of Auto-correlation within error terms
————————————————————————————————————————————————————————
Now, we fit a distributed lag model for ASX price and Gold Price (AUD)
asx_gold = data.frame(asx_data$`ASX price`,asx_data$`Gold price`)
asx_gold.ts = ts(asx_gold, start = 2004, frequency = 12)
head(asx_gold.ts)## asx_data..ASX.price. asx_data..Gold.price.
## Jan 2004 2935.4 611.9
## Feb 2004 2778.4 603.3
## Mar 2004 2848.6 565.7
## Apr 2004 2970.9 538.6
## May 2004 2979.8 549.4
## Jun 2004 2999.7 535.9asx_gold_scaled = scale(asx_gold.ts)
plot(asx_gold_scaled, plot.type = "s", col = c("blue","red"), main = "Fig 50. Time series Plot of Gold price (AUD) vs ASX Price Index")
legend("topleft", colnames(asx_gold_scaled), col=1:ncol(asx_gold), lty=1, cex=.70)
Distrubuted Lag Model of ASX price index and copper_usd/tonne with lag 1
model1g <- dlm(x = as.vector(gold), y = as.vector(asx),
q = 1)
summary(model1g)##
## Call:
## lm(formula = model.formula, data = design)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1654.13 -706.62 -9.17 561.27 2219.74
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3943.0338 211.5468 18.639 <2e-16 ***
## x.t 0.4117 1.2742 0.323 0.747
## x.1 0.3216 1.2711 0.253 0.801
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 839.7 on 157 degrees of freedom
## Multiple R-squared: 0.1097, Adjusted R-squared: 0.09832
## F-statistic: 9.669 on 2 and 157 DF, p-value: 0.0001097
##
## AIC and BIC values for the model:
## AIC BIC
## 1 2613.609 2625.91From the above model, we observe the following
Residual standard error = 839.7 on 157 degrees of freedom
The model is significant as the p_value is less 5% level of significance
F-statistic of the model = 9.669
The multiple and adjusted R_sqrd are 10.97 and 9.832 respectively. However the adjusted R_sqrd is very small
Also, no variables except the intercept in the model are significant as they are greater than 5% level of significance
Residual check and Serial Correlation Check using Breusch-Godfrey
checkresiduals(model1g$model)
##
## Breusch-Godfrey test for serial correlation of order up to 10
##
## data: Residuals
## LM test = 148.3, df = 10, p-value < 2.2e-16From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, p_value of breusch-godfrey test is less that 5% level of significance. This concludes serial correlation in the residuals is highly significant.
Multi-collinearity Check
vif_model1g <- vif(model1g$model)
vif_model1g## x.t x.1
## 58.50301 58.50301vif_model1g > 10## x.t x.1
## TRUE TRUEAs the Variation inflation factor is greater than 10, we conclude the effect of multi-collinearity is high
Specification of lag length
model1g_AIC_lag_length <- finiteDLMauto(x = as.vector(gold), y = as.vector(asx),
q.min = 1, q.max = 12, model.type = "dlm",
error.type = "AIC", trace = TRUE)
model1g_AIC_lag_length## q - k MASE AIC BIC GMRAE MBRAE R.Adj.Sq Ljung-Box
## 12 12 3.78532 2425.883 2470.942 4.08977 1.42891 -0.04664 0
## 11 11 3.85987 2442.989 2485.138 4.27351 1.04343 -0.03486 0
## 10 10 3.91844 2460.345 2499.570 4.32190 0.69116 -0.02199 0
## 9 9 3.97365 2476.983 2513.270 4.43213 0.75893 -0.00887 0
## 8 8 4.02150 2493.885 2527.220 4.45929 0.61156 0.00491 0
## 7 7 4.07437 2510.535 2540.905 4.47998 0.92888 0.01807 0
## 6 6 4.11652 2527.575 2554.966 4.40970 1.22132 0.03067 0
## 5 5 4.16217 2544.887 2569.286 4.45961 0.93030 0.04375 0
## 4 4 4.22171 2562.296 2583.690 4.65274 1.30422 0.05548 0
## 3 3 4.27118 2579.215 2597.590 4.64464 1.61370 0.06908 0
## 2 2 4.31239 2596.292 2611.637 4.68194 2.08243 0.08409 0
## 1 1 4.35631 2613.609 2625.910 4.56013 1.06887 0.09832 0model1g_BIC_lag_length <-
finiteDLMauto(x =as.vector(gold), y = as.vector(asx),
q.min = 1, q.max = 12, error.type = "BIC",
model.type = "dlm", trace = TRUE)
model1g_BIC_lag_length## q - k MASE AIC BIC GMRAE MBRAE R.Adj.Sq Ljung-Box
## 12 12 3.78532 2425.883 2470.942 4.08977 1.42891 -0.04664 0
## 11 11 3.85987 2442.989 2485.138 4.27351 1.04343 -0.03486 0
## 10 10 3.91844 2460.345 2499.570 4.32190 0.69116 -0.02199 0
## 9 9 3.97365 2476.983 2513.270 4.43213 0.75893 -0.00887 0
## 8 8 4.02150 2493.885 2527.220 4.45929 0.61156 0.00491 0
## 7 7 4.07437 2510.535 2540.905 4.47998 0.92888 0.01807 0
## 6 6 4.11652 2527.575 2554.966 4.40970 1.22132 0.03067 0
## 5 5 4.16217 2544.887 2569.286 4.45961 0.93030 0.04375 0
## 4 4 4.22171 2562.296 2583.690 4.65274 1.30422 0.05548 0
## 3 3 4.27118 2579.215 2597.590 4.64464 1.61370 0.06908 0
## 2 2 4.31239 2596.292 2611.637 4.68194 2.08243 0.08409 0
## 1 1 4.35631 2613.609 2625.910 4.56013 1.06887 0.09832 0We see from the AIC and BIC criteria that lag 12 has the lowest AIC and BIC values. Hence, we specify the lag length as 12.
model1.1g <- dlm(x = as.vector(gold), y = as.vector(asx), q = 12)
summary(model1.1g)##
## Call:
## lm(formula = model.formula, data = design)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1550.43 -505.36 30.42 500.55 1884.16
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4683.75496 230.73258 20.299 <2e-16 ***
## x.t -0.73063 1.27183 -0.574 0.567
## x.1 -0.05830 1.88620 -0.031 0.975
## x.2 0.06822 1.90429 0.036 0.971
## x.3 0.02231 1.91389 0.012 0.991
## x.4 -0.38688 1.90638 -0.203 0.839
## x.5 0.37549 1.90912 0.197 0.844
## x.6 0.08537 1.93160 0.044 0.965
## x.7 0.61146 1.93935 0.315 0.753
## x.8 -0.28852 1.95099 -0.148 0.883
## x.9 0.17788 1.96992 0.090 0.928
## x.10 0.22900 1.95773 0.117 0.907
## x.11 -0.06205 1.94273 -0.032 0.975
## x.12 0.22419 1.29372 0.173 0.863
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 788.6 on 135 degrees of freedom
## Multiple R-squared: 0.04529, Adjusted R-squared: -0.04664
## F-statistic: 0.4926 on 13 and 135 DF, p-value: 0.9257
##
## AIC and BIC values for the model:
## AIC BIC
## 1 2425.883 2470.942From the above model, we observe the following
Residual standard error = 788.6 on 135 degrees of freedom
The model is not significant as the p_value is greater than 5% level of significance
F-statistic of the model = 0.4926
The multiple and adjusted R_sqrd are 4.529 and -4.664 respectively. The adjusted R_sqrd is very small
Also, no variables except the intercept in the model are significant as they are greater than 5% level of significance
Residual check and Serial Correlation Check using Breusch-Godfrey
checkresiduals(model1.1g$model)
##
## Breusch-Godfrey test for serial correlation of order up to 17
##
## data: Residuals
## LM test = 136.44, df = 17, p-value < 2.2e-16From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, p_value of breusch-godfrey test is less that 5% level of significance. This concludes serial correlation in the residuals is highly significant.
Multi-collinearity Check
vif_model1.1g <- vif(model1.1g$model)
vif_model1.1g## x.t x.1 x.2 x.3 x.4 x.5 x.6 x.7
## 53.00292 118.33830 122.33823 125.52489 126.19786 128.35765 133.47622 136.12418
## x.8 x.9 x.10 x.11 x.12
## 139.03106 142.47802 141.04707 138.69245 61.50449vif_model1.1g > 10## x.t x.1 x.2 x.3 x.4 x.5 x.6 x.7 x.8 x.9 x.10 x.11 x.12
## TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUEAs the Variation inflation factor is greater than 10, we conclude the effect of multi-collinearity is high
Polynomial Distributed Lag Model
We can reduce the effect of multi-collinearity using the polynomial distributed lag model
model2g <- polyDlm(x = as.vector(gold), y = as.vector(asx),
q = 1, k = 1, show.beta = FALSE)
summary(model2g)##
## Call:
## "Y ~ (Intercept) + X.t"
##
## Residuals:
## Min 1Q Median 3Q Max
## -1654.13 -706.62 -9.17 561.27 2219.74
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3943.03376 211.54676 18.639 <2e-16 ***
## z.t0 0.41169 1.27421 0.323 0.747
## z.t1 -0.09005 2.53981 -0.035 0.972
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 839.7 on 157 degrees of freedom
## Multiple R-squared: 0.1097, Adjusted R-squared: 0.09832
## F-statistic: 9.669 on 2 and 157 DF, p-value: 0.0001097From the above model, we observe the following
Residual standard error = 839.7 on 157 degrees of freedom
The model is significant as the p_value is less 5% level of significance
F-statistic of the model = 9.669
The multiple and adjusted R_sqrd are 10.97 and 9.832 respectively. However the adjusted R_sqrd is very small
Also, no variables except the intercept in the model are significant as they are greater than 5% level of significance
A polynomial pattern (q = 4, k = 4) is imposed on the lag distribution i.e. equivalent to represent beta parameters with Kth order polynomial model time.
model2.2g <- polyDlm(x = as.vector(gold), y = as.vector(asx),
q = 4, k = 4, show.beta = TRUE)## Estimates and t-tests for beta coefficients:
## Estimate Std. Error t value P(>|t|)
## beta.0 0.0328 1.28 0.0256 0.980
## beta.1 -0.1190 1.92 -0.0621 0.951
## beta.2 0.3650 1.95 0.1870 0.852
## beta.3 -0.3040 1.93 -0.1570 0.875
## beta.4 0.6500 1.28 0.5090 0.611summary(model2.2g)##
## Call:
## "Y ~ (Intercept) + X.t"
##
## Residuals:
## Min 1Q Median 3Q Max
## -1574.26 -685.10 -6.82 528.30 2168.22
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4118.08645 214.92803 19.160 <2e-16 ***
## z.t0 0.03279 1.28010 0.026 0.980
## z.t1 -2.20885 15.15710 -0.146 0.884
## z.t2 3.30652 19.49855 0.170 0.866
## z.t3 -1.44009 7.84269 -0.184 0.855
## z.t4 0.19029 0.97843 0.194 0.846
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 825.6 on 151 degrees of freedom
## Multiple R-squared: 0.08576, Adjusted R-squared: 0.05548
## F-statistic: 2.833 on 5 and 151 DF, p-value: 0.01784In the above polynomial distributed lag model, we observe that all the variables are not significant to 5% level of significance
The residual standard error = 825.6 on 151 degree of freedom
The multiple and adjusted R-sqrd are 8.576 and 5.548 respectively
The F-statistic is 2.833 on 5 and 151 degrees of freedom
Residual check and Serial Correlation Check using Breusch-Godfrey
checkresiduals(model2.2g$model)
##
## Breusch-Godfrey test for serial correlation of order up to 10
##
## data: Residuals
## LM test = 145.02, df = 10, p-value < 2.2e-16From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, p_value of breusch-godfrey test is less that 5% level of significance. This concludes serial correlation in the residuals is highly significant.
Multi-collinearity Check
vif_model2.2g <- vif(model2.2g$model)
vif_model2.2g## z.t0 z.t1 z.t2 z.t3 z.t4
## 1454.893 826492.983 12368348.542 22307303.143 4361029.001vif_model2.2g > 10## z.t0 z.t1 z.t2 z.t3 z.t4
## TRUE TRUE TRUE TRUE TRUEAs the Variation inflation factor is greater than 10, we conclude the effect of multi-collinearity is high
Koyck Distributed Lag Model
model3g <- koyckDlm(x = as.vector(gold), y = as.vector(asx))
summary(model3g, diagnostics = TRUE)##
## Call:
## "Y ~ (Intercept) + Y.1 + X.t"
##
## Residuals:
## Min 1Q Median 3Q Max
## -682.19 -105.44 15.86 135.04 783.60
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.902e+02 8.958e+01 2.123 0.0353 *
## Y.1 9.635e-01 1.909e-02 50.469 <2e-16 ***
## X.t 2.595e-03 4.304e-02 0.060 0.9520
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 201.4 on 157 degrees of freedom
## Multiple R-Squared: 0.9488, Adjusted R-squared: 0.9481
## Wald test: 1454 on 2 and 157 DF, p-value: < 2.2e-16
##
## Diagnostic tests:
## df1 df2 statistic p-value
## Weak instruments 1 157 8006.63657 1.266826e-136
## Wu-Hausman 1 156 18.06854 3.655601e-05
##
## alpha beta phi
## Geometric coefficients: 5205.15 0.002595168 0.9634602From the above model, we observe the following
Residual standard error = 201.4 on 157 degrees of freedom
The model is significant as the p_value is less 5% level of significance
The multiple and adjusted R_sqrd are 94.88 and 94.81 respectively. The adjusted R_sqrd is comparitively greater than the Distributed Lag & Polynomial Distributed Lag model
Also, variable Y.1 in the model is significant to 5% level of significance. The beta and phi values are 0.0026 and 0.963 respectively
The weak instruments p_value suggests that the model is significant at first stage of least squares fitting
Wu-Hausman test is used to compare the consistency of instrumental variable estimates to ordinary least square estimates. The presence of endogeniety tells us that instrumental variable estimate is consistent and ordinary least square estimate is inconsistent.
The Wu-Hausman test is significant as the p_value is less than 5% level of significance. Hence, we reject the null hypothesis that correlation between explanatory vairable and error term is zero or in other words endogeniety is not present.
Therefore, we conclude that there is significant correlation between explanatory variable and error term is present at 5% level of significance.
Residual Check
checkresiduals(model3g$model)
From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Multi-collinearity Check
vif_model3g <- vif(model3g$model)
vif_model3g## Y.1 X.t
## 1.140648 1.140648vif_model3g > 10## Y.1 X.t
## FALSE FALSEThe variation inflation factor is less than 5 when value of q = 2, we conclude the effect of multi-collinearity is not present.
Auto Regressive Distributed Lag Model
model4g <- ardlDlm(x = as.vector(gold), y = as.vector(asx),
p = 1, q =1)
summary(model4g)##
## Time series regression with "ts" data:
## Start = 2, End = 161
##
## Call:
## dynlm(formula = as.formula(model.text), data = data, start = 1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -598.9 -102.9 10.4 119.5 724.6
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 182.80452 85.05812 2.149 0.0332 *
## X.t -1.24911 0.29162 -4.283 3.21e-05 ***
## X.1 1.23169 0.28976 4.251 3.66e-05 ***
## Y.1 0.97172 0.01812 53.624 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 191.1 on 156 degrees of freedom
## Multiple R-squared: 0.9542, Adjusted R-squared: 0.9533
## F-statistic: 1083 on 3 and 156 DF, p-value: < 2.2e-16From the above model, we observe the following
Residual standard error = 191.1 on 156 degrees of freedom
The model is significant as the p_value is less 5% level of significance
F-statistic of the model = 1083 on 3 and 156 Degrees of freedom
The multiple and adjusted R_sqrd are 95.42 and 95.33 respectively. The adjusted R_sqrd of this model is comparitively higher than DLM, Polynomial DLM and Koyck models.
Also, all the variables in the model are significant as they are smaller than 5% level of significance
Residuals Check
checkresiduals(model4g$model)
##
## Breusch-Godfrey test for serial correlation of order up to 10
##
## data: Residuals
## LM test = 7.9182, df = 10, p-value = 0.6368The breusch-godfrey test has a p_value greater than 5% level of significance. Hence, it states we reject the null hypothesis that variance is not changing in the residuals.
From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, we can infer that the standardized residuals in the histogram are distributed around zero mean levels
The ACF plot of the residuals does not suggest the existence of Auto-correlation within error terms
Multi-collinearity Check
vif_model4g <- vif(model4g$model)
vif_model4g## X.t L(X.t, 1) L(y.t, 1)
## 59.17040 58.70438 1.14186vif_model4g > 10## X.t L(X.t, 1) L(y.t, 1)
## TRUE TRUE FALSEThe variation inflation factor is less than 5 for coefficient Y.t when value of q = 1.
Determining the values of p and q using AIC and BIC values
for(i in 1:5){
for(j in 1:5){
model4.1g <- ardlDlm(x = as.vector(gold),
y = as.vector(asx), p = i, q = j)
cat("p:",i,"q:",j,"AIC:",AIC(model4.1g$model), "BIC:",
BIC(model4.1g$model),"\n")
}
}## p: 1 q: 1 AIC: 2140.897 BIC: 2156.273
## p: 1 q: 2 AIC: 2128.524 BIC: 2146.938
## p: 1 q: 3 AIC: 2113.99 BIC: 2135.428
## p: 1 q: 4 AIC: 2102.754 BIC: 2127.204
## p: 1 q: 5 AIC: 2092.194 BIC: 2119.643
## p: 2 q: 1 AIC: 2128.627 BIC: 2147.04
## p: 2 q: 2 AIC: 2130.523 BIC: 2152.005
## p: 2 q: 3 AIC: 2115.89 BIC: 2140.39
## p: 2 q: 4 AIC: 2104.694 BIC: 2132.2
## p: 2 q: 5 AIC: 2094.14 BIC: 2124.639
## p: 3 q: 1 AIC: 2118.109 BIC: 2139.547
## p: 3 q: 2 AIC: 2120.027 BIC: 2144.528
## p: 3 q: 3 AIC: 2117.305 BIC: 2144.868
## p: 3 q: 4 AIC: 2105.731 BIC: 2136.293
## p: 3 q: 5 AIC: 2095.264 BIC: 2128.812
## p: 4 q: 1 AIC: 2107.002 BIC: 2131.452
## p: 4 q: 2 AIC: 2108.914 BIC: 2136.42
## p: 4 q: 3 AIC: 2106.276 BIC: 2136.839
## p: 4 q: 4 AIC: 2107.456 BIC: 2141.074
## p: 4 q: 5 AIC: 2097.01 BIC: 2133.608
## p: 5 q: 1 AIC: 2094.908 BIC: 2122.357
## p: 5 q: 2 AIC: 2096.86 BIC: 2127.359
## p: 5 q: 3 AIC: 2094.144 BIC: 2127.692
## p: 5 q: 4 AIC: 2095.425 BIC: 2132.023
## p: 5 q: 5 AIC: 2097.324 BIC: 2136.972We observe that p = 1 and q = 5 give the smallest AIC and BIC values.
best_model_gold <- ardlDlm(x = as.vector(gold), y = as.vector(asx),
p = 1, q = 5)
summary(best_model_gold)##
## 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
## -564.02 -106.74 8.99 126.34 691.74
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 234.72597 92.97346 2.525 0.012635 *
## X.t -1.21331 0.30480 -3.981 0.000107 ***
## X.1 1.21097 0.30013 4.035 8.73e-05 ***
## Y.1 0.96620 0.07927 12.189 < 2e-16 ***
## Y.2 0.13687 0.11316 1.210 0.228368
## Y.3 -0.07572 0.11193 -0.676 0.499816
## Y.4 -0.04931 0.11174 -0.441 0.659640
## Y.5 -0.02130 0.07871 -0.271 0.787092
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 191.6 on 148 degrees of freedom
## Multiple R-squared: 0.9502, Adjusted R-squared: 0.9478
## F-statistic: 403.1 on 7 and 148 DF, p-value: < 2.2e-16From the above model, we observe the following
Residual standard error = 191.6 on 148 degrees of freedom
The model is significant as the p_value is less 5% level of significance
F-statistic of the model = 403.1 on 7 and 148 Degrees of freedom
The multiple and adjusted R_sqrd are 95.02 and 94.78 respectively. The adjusted R_sqrd of this model is comparitively higher.
Variables X.t, X.1 and Y.1 in the model are significant as the p_value is smaller than 5% level of significance
Residuals Check
checkresiduals(best_model_gold$model)
##
## Breusch-Godfrey test for serial correlation of order up to 11
##
## data: Residuals
## LM test = 5.7145, df = 11, p-value = 0.8917The breusch-godfrey test has a p_value greater than 5% level of significance. Hence, it states we reject the null hypothesis that variance is not changing in the residuals.
From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, we can infer that the standardized residuals in the histogram are distributed around zero mean levels
The ACF plot of the residuals do not suggest the existence of Auto-correlation within error terms
#————————————————————————————————————————————————————————-
Now, we fit a distributed lag model for ASX price and Crude Oil
asx_oil = data.frame(asx_data$`ASX price`, asx_data$`Crude Oil (Brent)_USD/bbl`)
asx_oil.ts = ts(asx_oil, start = 2004, frequency = 12)
head(asx_oil.ts)## asx_data..ASX.price. asx_data..Crude.Oil..Brent._USD.bbl.
## Jan 2004 2935.4 31.29
## Feb 2004 2778.4 32.65
## Mar 2004 2848.6 30.34
## Apr 2004 2970.9 25.02
## May 2004 2979.8 25.81
## Jun 2004 2999.7 27.55asx_oil_scaled = scale(asx_oil.ts)
plot(asx_oil_scaled, plot.type = "s", col = c("blue","red"),main = "Fig 51. Time series Plot of Crude oil price (USD/litre) vs ASX Price Index")
legend("bottomright", colnames(asx_oil.ts), col=1:ncol(asx_oil), lty=2, cex=.75)
Distrubuted Lag Model of ASX price index and Crude Oil with lag 1
model1crude <- dlm(x = as.vector(oil), y = as.vector(asx),
q = 1)
summary(model1crude)##
## Call:
## lm(formula = model.formula, data = design)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1677.08 -770.94 -61.63 694.56 1823.95
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4144.338 178.033 23.278 <2e-16 ***
## x.t 17.140 10.636 1.611 0.109
## x.1 -7.941 10.589 -0.750 0.454
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 842.6 on 157 degrees of freedom
## Multiple R-squared: 0.1036, Adjusted R-squared: 0.09217
## F-statistic: 9.071 on 2 and 157 DF, p-value: 0.000187
##
## AIC and BIC values for the model:
## AIC BIC
## 1 2614.698 2626.998From the above model, we observe the following
Residual standard error = 842.6 on 157 degrees of freedom
The model is significant as the p_value is less 5% level of significance
F-statistic of the model = 9.071
The multiple and adjusted R_sqrd are 10.36 and 9.217 respectively. However the adjusted R_sqrd is very small
Also, no variables except the intercept in the model are significant as they are greater than 5% level of significance
Residual check and Serial Correlation Check using Breusch-Godfrey
checkresiduals(model1crude$model)
##
## Breusch-Godfrey test for serial correlation of order up to 10
##
## data: Residuals
## LM test = 147.28, df = 10, p-value < 2.2e-16From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, p_value of breusch-godfrey test is less that 5% level of significance. This concludes serial correlation in the residuals is highly significant.
Multi-collinearity Check
vif_model1crude <- vif(model1crude$model)
vif_model1crude## x.t x.1
## 22.79751 22.79751vif_model1crude > 10## x.t x.1
## TRUE TRUEAs the Variation inflation factor is greater than 10, we conclude the effect of multi-collinearity is high
Specification of lag length:
model1crude_AIC_lag_length <- finiteDLMauto(x = as.vector(oil), y = as.vector(asx),
q.min = 1, q.max = 12, model.type = "dlm",
error.type = "AIC", trace = TRUE)
model1crude_AIC_lag_length## q - k MASE AIC BIC GMRAE MBRAE R.Adj.Sq Ljung-Box
## 12 12 3.85940 2424.123 2469.182 4.49414 1.18212 -0.03435 0
## 11 11 3.94102 2442.030 2484.178 4.50351 1.71454 -0.02826 0
## 10 10 4.00878 2459.842 2499.066 4.39830 0.94829 -0.01859 0
## 9 9 4.07657 2476.871 2513.158 4.59976 0.86566 -0.00812 0
## 8 8 4.13953 2493.914 2527.249 4.76106 0.78221 0.00472 0
## 7 7 4.20880 2510.754 2541.124 4.79551 0.02285 0.01667 0
## 6 6 4.26700 2527.701 2555.091 4.84721 0.56283 0.02988 0
## 5 5 4.32339 2544.936 2569.335 4.87419 1.47404 0.04345 0
## 4 4 4.39825 2561.888 2583.281 5.11093 1.10612 0.05794 0
## 3 3 4.47189 2579.101 2597.477 5.24099 1.43485 0.06975 0
## 2 2 4.53543 2596.715 2612.059 5.37682 0.74046 0.08165 0
## 1 1 4.60656 2614.698 2626.998 5.46829 0.63464 0.09217 0model1crude_BIC_lag_length <-
finiteDLMauto(x =as.vector(oil), y = as.vector(asx),
q.min = 1, q.max = 12, error.type = "BIC",
model.type = "dlm", trace = TRUE)
model1crude_BIC_lag_length## q - k MASE AIC BIC GMRAE MBRAE R.Adj.Sq Ljung-Box
## 12 12 3.85940 2424.123 2469.182 4.49414 1.18212 -0.03435 0
## 11 11 3.94102 2442.030 2484.178 4.50351 1.71454 -0.02826 0
## 10 10 4.00878 2459.842 2499.066 4.39830 0.94829 -0.01859 0
## 9 9 4.07657 2476.871 2513.158 4.59976 0.86566 -0.00812 0
## 8 8 4.13953 2493.914 2527.249 4.76106 0.78221 0.00472 0
## 7 7 4.20880 2510.754 2541.124 4.79551 0.02285 0.01667 0
## 6 6 4.26700 2527.701 2555.091 4.84721 0.56283 0.02988 0
## 5 5 4.32339 2544.936 2569.335 4.87419 1.47404 0.04345 0
## 4 4 4.39825 2561.888 2583.281 5.11093 1.10612 0.05794 0
## 3 3 4.47189 2579.101 2597.477 5.24099 1.43485 0.06975 0
## 2 2 4.53543 2596.715 2612.059 5.37682 0.74046 0.08165 0
## 1 1 4.60656 2614.698 2626.998 5.46829 0.63464 0.09217 0We see from the AIC and BIC criteria that lag 12 has the lowest AIC and BIC values. Hence, we specify the lag length as 12.
model1.1crude <- dlm(x = as.vector(oil), y = as.vector(asx), q = 12)
summary(model1.1crude)##
## Call:
## lm(formula = model.formula, data = design)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1603.38 -588.61 59.47 510.06 1586.31
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4797.8928 203.0857 23.625 <2e-16 ***
## x.t 8.7184 11.1651 0.781 0.436
## x.1 0.9094 18.7149 0.049 0.961
## x.2 -0.7991 18.9816 -0.042 0.966
## x.3 0.7995 19.0667 0.042 0.967
## x.4 -4.3798 19.0414 -0.230 0.818
## x.5 0.2400 19.1322 0.013 0.990
## x.6 1.0663 19.2530 0.055 0.956
## x.7 -5.0585 19.4123 -0.261 0.795
## x.8 0.9627 19.3634 0.050 0.960
## x.9 -2.0805 19.3391 -0.108 0.914
## x.10 1.0344 19.3332 0.054 0.957
## x.11 -8.1813 19.0500 -0.429 0.668
## x.12 8.7236 11.1989 0.779 0.437
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 784 on 135 degrees of freedom
## Multiple R-squared: 0.0565, Adjusted R-squared: -0.03435
## F-statistic: 0.6219 on 13 and 135 DF, p-value: 0.8325
##
## AIC and BIC values for the model:
## AIC BIC
## 1 2424.123 2469.182From the above model, we observe the following
Residual standard error = 784 on 135 degrees of freedom
The model is not significant as the p_value is greater than 5% level of significance
F-statistic of the model = 0.6219
The multiple and adjusted R_sqrd are 5.65 and 3.435 respectively. However the adjusted R_sqrd is very small
Also, no variables except the intercept in the model are significant as they are greater than 5% level of significance
Residual check and Serial Correlation Check using Breusch-Godfrey
checkresiduals(model1.1crude$model)
##
## Breusch-Godfrey test for serial correlation of order up to 17
##
## data: Residuals
## LM test = 135.49, df = 17, p-value < 2.2e-16From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, p_value of breusch-godfrey test is less that 5% level of significance. This concludes serial correlation in the residuals is highly significant.
Multi-collinearity Check
vif_model1.1crude <- vif(model1.1crude$model)
vif_model1.1crude## x.t x.1 x.2 x.3 x.4 x.5 x.6 x.7
## 24.09715 68.58448 71.58093 73.18015 74.17098 75.89967 77.94797 80.15117
## x.8 x.9 x.10 x.11 x.12
## 80.86523 81.70058 82.37142 80.51631 28.07618vif_model1.1crude > 10## x.t x.1 x.2 x.3 x.4 x.5 x.6 x.7 x.8 x.9 x.10 x.11 x.12
## TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUEAs the Variation inflation factor is greater than 10, we conclude the effect of multi-collinearity is high
Polynomial Distributed Lag Model
We can reduce the effect of multi-collinearity using the polynomial distributed lag model
model2crude <- polyDlm(x = as.vector(oil), y = as.vector(asx),
q = 1, k = 1, show.beta = FALSE)
summary(model2crude)##
## Call:
## "Y ~ (Intercept) + X.t"
##
## Residuals:
## Min 1Q Median 3Q Max
## -1677.08 -770.94 -61.63 694.56 1823.95
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4144.34 178.03 23.278 <2e-16 ***
## z.t0 17.14 10.64 1.611 0.109
## z.t1 -25.08 21.11 -1.188 0.237
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 842.6 on 157 degrees of freedom
## Multiple R-squared: 0.1036, Adjusted R-squared: 0.09217
## F-statistic: 9.071 on 2 and 157 DF, p-value: 0.000187From the above model, we observe the following
Residual standard error = 842.6 on 157 degrees of freedom
The model is significant as the p_value is less 5% level of significance
F-statistic of the model = 9.071
The multiple and adjusted R_sqrd are 10.36 and 9.217 respectively. However the adjusted R_sqrd is very small
Also, no variables except the intercept in the model are significant as they are greater than 5% level of significance
A polynomial pattern (q = 4, k = 4) is imposed on the lag distribution i.e. equivalent to represent beta parameters with Kth order polynomial model time.
model2.2crude <- polyDlm(x = as.vector(oil), y = as.vector(asx),
q = 4, k = 4, show.beta = TRUE)## Estimates and t-tests for beta coefficients:
## Estimate Std. Error t value P(>|t|)
## beta.0 10.50 11.5 0.9090 0.365
## beta.1 2.71 19.3 0.1400 0.889
## beta.2 -1.30 19.5 -0.0664 0.947
## beta.3 3.76 19.3 0.1940 0.846
## beta.4 -8.78 11.4 -0.7670 0.444In the above polynomial distributed lag model, we observe that all the variables are not significant to 5% level of significance.
Residual check and Serial Correlation Check using Breusch-Godfrey
checkresiduals(model2.2crude$model)
##
## Breusch-Godfrey test for serial correlation of order up to 10
##
## data: Residuals
## LM test = 145.13, df = 10, p-value < 2.2e-16From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, p_value of breusch-godfrey test is less that 5% level of significance. This concludes serial correlation in the residuals is highly significant.
Multi-collinearity Check
vif_model2.2crude <- vif(model2.2crude$model)
vif_model2.2crude## z.t0 z.t1 z.t2 z.t3 z.t4
## 6.500776e+02 4.348826e+05 6.573599e+06 1.189865e+07 2.330502e+06vif_model2.2crude > 10## z.t0 z.t1 z.t2 z.t3 z.t4
## TRUE TRUE TRUE TRUE TRUEAs the Variation inflation factor is greater than 10, we conclude the effect of multi-collinearity is high
Koyck Distributed Lag Model
model3crude <- koyckDlm(x = as.vector(oil), y = as.vector(asx))
summary(model3crude, diagnostics = TRUE)##
## Call:
## "Y ~ (Intercept) + Y.1 + X.t"
##
## Residuals:
## Min 1Q Median 3Q Max
## -683.91 -108.66 13.68 139.77 762.55
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 209.89536 87.89368 2.388 0.0181 *
## Y.1 0.97537 0.01905 51.193 <2e-16 ***
## X.t -0.99907 0.58045 -1.721 0.0872 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 201.1 on 157 degrees of freedom
## Multiple R-Squared: 0.949, Adjusted R-squared: 0.9483
## Wald test: 1461 on 2 and 157 DF, p-value: < 2.2e-16
##
## Diagnostic tests:
## df1 df2 statistic p-value
## Weak instruments 1 157 3018.4519 1.997814e-104
## Wu-Hausman 1 156 11.1979 1.026113e-03
##
## alpha beta phi
## Geometric coefficients: 8522.034 -0.9990694 0.9753703From the above model, we observe the following
Residual standard error = 201.1 on 157 degrees of freedom
The model is significant as the p_value is less 5% level of significance
The multiple and adjusted R_sqrd are 94.9 and 94.83 respectively. The adjusted R_sqrd are comparitively greater than the Distributed Lag & Polynomial Distributed Lag model
Also, variable Y.1 in the model is significant to 5% level of significance. The beta and phi values are -0.9990694 and 0.9753703 respectively
The weak instruments p_value suggests that the model is significant at first stage of least squares fitting
Wu-Hausman test is used to compare the consistency of instrumental variable estimates to ordinary least square estimates. The presence of endogeniety tells us that instrumental variable estimate is consistent and ordinary least square estimate is inconsistent.
The Wu-Hausman test is significant as the p_value is less than 5% level of significance. Hence, we reject the null hypothesis that correlation between explanatory vairable and error term is zero or in other words endogeniety is not present.
Therefore, we conclude that there is significant correlation between explanatory variable and error term is present at 5% level of significance.
Residual Check
checkresiduals(model3crude$model)
From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Multi-collinearity Check
vif_model3crude <- vif(model3crude$model)
vif_model3crude## Y.1 X.t
## 1.14038 1.14038vif_model3crude > 10## Y.1 X.t
## FALSE FALSEThe variation inflation factor is less than 5 when value of q = 2, we conclude the effect of multi-collinearity is not present.
Auto Regressive Distributed Lag Model
model4crude <- ardlDlm(x = as.vector(oil), y = as.vector(asx),
p = 1, q =1)
summary(model4crude)##
## Time series regression with "ts" data:
## Start = 2, End = 161
##
## Call:
## dynlm(formula = as.formula(model.text), data = data, start = 1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -684.06 -111.57 -1.77 138.90 719.35
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 211.45397 85.03705 2.487 0.01395 *
## X.t 7.45111 2.46199 3.026 0.00290 **
## X.1 -8.17917 2.44422 -3.346 0.00103 **
## Y.1 0.97067 0.01837 52.827 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 194.5 on 156 degrees of freedom
## Multiple R-squared: 0.9525, Adjusted R-squared: 0.9516
## F-statistic: 1044 on 3 and 156 DF, p-value: < 2.2e-16From the above model, we observe the following
Residual standard error = 194.5 on 156 degrees of freedom
The model is significant as the p_value is less 5% level of significance
F-statistic of the model = 1044 on 3 and 156 Degrees of freedom
The multiple and adjusted R_sqrd are 95.25 and 95.16 respectively. The adjusted R_sqrd of this model is comparitively higher than DLM, Polynomial DLM and Koyck models.
Also, all the variables in the model are significant as they are smaller than 5% level of significance
Residuals Check
checkresiduals(model4crude$model)
##
## Breusch-Godfrey test for serial correlation of order up to 10
##
## data: Residuals
## LM test = 6.0252, df = 10, p-value = 0.8131The breusch-godfrey test has a p_value greater than 5% level of significance. Hence, it states we reject the null hypothesis that variance is not changing in the residuals.
From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, we can infer that the standardized residuals in the histogram are distributed around zero mean levels
The ACF plot of the residuals do not suggest the existence of Auto-correlation within error terms
Multi-collinearity Check
vif_model4crude <- vif(model4crude$model)
vif_model4crude## X.t L(X.t, 1) L(y.t, 1)
## 22.924733 22.797583 1.133443vif_model4crude > 10## X.t L(X.t, 1) L(y.t, 1)
## TRUE TRUE FALSEThe variation inflation factor is less than 5 for coefficient Y.t when value of q = 1.
Determining the values of p and q using AIC and BIC values
for(i in 1:5){
for(j in 1:5){
model4.1crude <- ardlDlm(x = as.vector(oil),
y = as.vector(asx), p = i, q = j)
cat("p:",i,"q:",j,"AIC:",AIC(model4.1crude$model), "BIC:",
BIC(model4.1crude$model),"\n")
}
}## p: 1 q: 1 AIC: 2146.524 BIC: 2161.9
## p: 1 q: 2 AIC: 2134.107 BIC: 2152.521
## p: 1 q: 3 AIC: 2121.07 BIC: 2142.508
## p: 1 q: 4 AIC: 2109.4 BIC: 2133.85
## p: 1 q: 5 AIC: 2098.335 BIC: 2125.784
## p: 2 q: 1 AIC: 2132.312 BIC: 2150.726
## p: 2 q: 2 AIC: 2134.235 BIC: 2155.718
## p: 2 q: 3 AIC: 2122.356 BIC: 2146.857
## p: 2 q: 4 AIC: 2110.793 BIC: 2138.299
## p: 2 q: 5 AIC: 2099.752 BIC: 2130.251
## p: 3 q: 1 AIC: 2121.919 BIC: 2143.357
## p: 3 q: 2 AIC: 2123.835 BIC: 2148.335
## p: 3 q: 3 AIC: 2124.324 BIC: 2151.887
## p: 3 q: 4 AIC: 2112.401 BIC: 2142.963
## p: 3 q: 5 AIC: 2101.35 BIC: 2134.899
## p: 4 q: 1 AIC: 2111.383 BIC: 2135.832
## p: 4 q: 2 AIC: 2113.294 BIC: 2140.8
## p: 4 q: 3 AIC: 2113.805 BIC: 2144.367
## p: 4 q: 4 AIC: 2114.384 BIC: 2148.003
## p: 4 q: 5 AIC: 2103.342 BIC: 2139.94
## p: 5 q: 1 AIC: 2097.076 BIC: 2124.525
## p: 5 q: 2 AIC: 2099.041 BIC: 2129.54
## p: 5 q: 3 AIC: 2099.518 BIC: 2133.066
## p: 5 q: 4 AIC: 2099.845 BIC: 2136.443
## p: 5 q: 5 AIC: 2100.917 BIC: 2140.566We observe that p = 5 and q = 1 give the smallest AIC and BIC values.
best_model_crude <- ardlDlm(x = as.vector(oil), y = as.vector(asx),
p = 5, q = 1)
summary(best_model_crude)##
## 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
## -657.99 -121.05 -4.38 123.84 714.56
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 239.54102 94.36245 2.539 0.0122 *
## X.t 5.68288 2.73874 2.075 0.0397 *
## X.1 -3.32354 4.57294 -0.727 0.4685
## X.2 -3.29907 4.62372 -0.714 0.4767
## X.3 3.05569 4.63039 0.660 0.5103
## X.4 -8.00745 4.57207 -1.751 0.0820 .
## X.5 5.08983 2.72010 1.871 0.0633 .
## Y.1 0.96648 0.01934 49.983 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 194.6 on 148 degrees of freedom
## Multiple R-squared: 0.9486, Adjusted R-squared: 0.9461
## F-statistic: 390 on 7 and 148 DF, p-value: < 2.2e-16From the above model, we observe the following
Residual standard error = 194.6 on 148 degrees of freedom
The model is significant as the p_value is less 5% level of significance
F-statistic of the model = 390 on 7 and 148 Degrees of freedom
The multiple and adjusted R_sqrd are 94.86 and 94.61 respectively. The adjusted R_sqrd of this model is comparitively higher than DLM, Polynomial DLM and Koyck models.
Variables X.t, X.1 and Y.1 in the model are significant as the p_value is smaller than 5% level of significance
Residuals Check
checkresiduals(best_model_crude$model)
##
## Breusch-Godfrey test for serial correlation of order up to 11
##
## data: Residuals
## LM test = 5.18, df = 11, p-value = 0.9221The breusch-godfrey test has a p_value greater than 5% level of significance. Hence, it states we reject the null hypothesis that variance is not changing in the residuals.
From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, we can infer that the standardized residuals in the histogram are distributed around zero mean levels
The ACF plot of the residuals do not suggest the existence of Auto-correlation within error terms
————————————————————————————————————————————————————
Now, we fit Distributed Lag model of ASX Price Index vs Gold + Copper + Crude_Oil
head(asx_data)## # A tibble: 6 x 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 1685asx_split1 = asx_price_index[,1]
head(asx_split1)## Jan Feb Mar Apr May Jun
## 2004 2935.4 2778.4 2848.6 2970.9 2979.8 2999.7asx_split2 = asx_price_index[,2:4]
head(asx_split2)## Gold price Crude Oil (Brent)_USD/bbl Copper_USD/tonne
## Jan 2004 611.9 31.29 1650
## Feb 2004 603.3 32.65 1682
## Mar 2004 565.7 30.34 1656
## Apr 2004 538.6 25.02 1588
## May 2004 549.4 25.81 1651
## Jun 2004 535.9 27.55 1685asx_all = data.frame(asx_split1,asx_split2)
asx_all.ts = ts(asx_all, start = 2004, frequency = 12)
head(asx_all.ts)## asx_split1 Gold.price Crude.Oil..Brent._USD.bbl Copper_USD.tonne
## Jan 2004 2935.4 611.9 31.29 1650
## Feb 2004 2778.4 603.3 32.65 1682
## Mar 2004 2848.6 565.7 30.34 1656
## Apr 2004 2970.9 538.6 25.02 1588
## May 2004 2979.8 549.4 25.81 1651
## Jun 2004 2999.7 535.9 27.55 1685asx_all_scaled = scale(asx_all.ts)
plot(asx_all_scaled, plot.type = "s", col = c("yellow","black","blue","red"), main = "Fig 52. Time series Plot")
legend("topleft",lty=1.25, text.width =1.25, col=c("blue", "yellow", "red", "black"), c("ASX", "Gold", "Oil", "Copper"))
Distrubuted Lag Model of ASX price index and copper_usd/tonne with lag 1
model1asx <- dlm(x = as.vector(asx_split2), y = as.vector(asx_split1),
q = 1)
summary(model1asx)##
## Call:
## lm(formula = model.formula, data = design)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1654.13 -706.62 -9.17 561.27 2219.74
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3943.0338 211.5468 18.639 <2e-16 ***
## x.t 0.4117 1.2742 0.323 0.747
## x.1 0.3216 1.2711 0.253 0.801
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 839.7 on 157 degrees of freedom
## (322 observations deleted due to missingness)
## Multiple R-squared: 0.1097, Adjusted R-squared: 0.09832
## F-statistic: 9.669 on 2 and 157 DF, p-value: 0.0001097
##
## AIC and BIC values for the model:
## AIC BIC
## 1 2613.609 2625.91From the above model, we observe the following
Residual standard error = 839.7 on 157 degrees of freedom
The model is significant as the p_value is less 5% level of significance
F-statistic of the model = 9.669
The multiple and adjusted R_sqrd are 10.97 and 9.832 respectively. However the adjusted R_sqrd is very small
Also, no variables except the intercept in the model are significant as they are greater than 5% level of significance
Residual check and Serial Correlation Check using Breusch-Godfrey
checkresiduals(model1asx$model)
##
## Breusch-Godfrey test for serial correlation of order up to 10
##
## data: Residuals
## LM test = 148.3, df = 10, p-value < 2.2e-16From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, p_value of breusch-godfrey test is less that 5% level of significance. This concludes serial correlation in the residuals is highly significant.
Multi-collinearity Check
vif_model1asx <- vif(model1asx$model)
vif_model1asx## x.t x.1
## 58.50301 58.50301vif_model1asx > 10## x.t x.1
## TRUE TRUEAs the Variation inflation factor is greater than 10, we conclude the effect of multi-collinearity is high
Specification of lag length:
model1asx_AIC_lag_length <- finiteDLMauto(x = as.vector(asx_split2), y = as.vector(asx_split1),
q.min = 1, q.max = 12, model.type = "dlm",
error.type = "AIC", trace = TRUE)
model1asx_AIC_lag_length## q - k MASE AIC BIC GMRAE MBRAE R.Adj.Sq Ljung-Box
## 12 12 3.78532 2425.883 2470.942 4.08977 1.42891 -0.04664 0
## 11 11 3.85987 2442.989 2485.138 4.27351 1.04343 -0.03486 0
## 10 10 3.91844 2460.345 2499.570 4.32190 0.69116 -0.02199 0
## 9 9 3.97365 2476.983 2513.270 4.43213 0.75893 -0.00887 0
## 8 8 4.02150 2493.885 2527.220 4.45929 0.61156 0.00491 0
## 7 7 4.07437 2510.535 2540.905 4.47998 0.92888 0.01807 0
## 6 6 4.11652 2527.575 2554.966 4.40970 1.22132 0.03067 0
## 5 5 4.16217 2544.887 2569.286 4.45961 0.93030 0.04375 0
## 4 4 4.22171 2562.296 2583.690 4.65274 1.30422 0.05548 0
## 3 3 4.27118 2579.215 2597.590 4.64464 1.61370 0.06908 0
## 2 2 4.31239 2596.292 2611.637 4.68194 2.08243 0.08409 0
## 1 1 4.35631 2613.609 2625.910 4.56013 1.06887 0.09832 0model1asx_BIC_lag_length <-
finiteDLMauto(x =as.vector(asx_split2), y = as.vector(asx_split1),
q.min = 1, q.max = 12, error.type = "BIC",
model.type = "dlm", trace = TRUE)
model1asx_BIC_lag_length## q - k MASE AIC BIC GMRAE MBRAE R.Adj.Sq Ljung-Box
## 12 12 3.78532 2425.883 2470.942 4.08977 1.42891 -0.04664 0
## 11 11 3.85987 2442.989 2485.138 4.27351 1.04343 -0.03486 0
## 10 10 3.91844 2460.345 2499.570 4.32190 0.69116 -0.02199 0
## 9 9 3.97365 2476.983 2513.270 4.43213 0.75893 -0.00887 0
## 8 8 4.02150 2493.885 2527.220 4.45929 0.61156 0.00491 0
## 7 7 4.07437 2510.535 2540.905 4.47998 0.92888 0.01807 0
## 6 6 4.11652 2527.575 2554.966 4.40970 1.22132 0.03067 0
## 5 5 4.16217 2544.887 2569.286 4.45961 0.93030 0.04375 0
## 4 4 4.22171 2562.296 2583.690 4.65274 1.30422 0.05548 0
## 3 3 4.27118 2579.215 2597.590 4.64464 1.61370 0.06908 0
## 2 2 4.31239 2596.292 2611.637 4.68194 2.08243 0.08409 0
## 1 1 4.35631 2613.609 2625.910 4.56013 1.06887 0.09832 0We see from the AIC and BIC criteria that lag 12 has the lowest AIC and BIC values. Hence, we specify the lag length as 12.
model1.1asx <- dlm(x = as.vector(asx_split2), y = as.vector(asx_split1), q = 12)
summary(model1.1asx)##
## Call:
## lm(formula = model.formula, data = design)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1550.43 -505.36 30.42 500.55 1884.16
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4683.75496 230.73258 20.299 <2e-16 ***
## x.t -0.73063 1.27183 -0.574 0.567
## x.1 -0.05830 1.88620 -0.031 0.975
## x.2 0.06822 1.90429 0.036 0.971
## x.3 0.02231 1.91389 0.012 0.991
## x.4 -0.38688 1.90638 -0.203 0.839
## x.5 0.37549 1.90912 0.197 0.844
## x.6 0.08537 1.93160 0.044 0.965
## x.7 0.61146 1.93935 0.315 0.753
## x.8 -0.28852 1.95099 -0.148 0.883
## x.9 0.17788 1.96992 0.090 0.928
## x.10 0.22900 1.95773 0.117 0.907
## x.11 -0.06205 1.94273 -0.032 0.975
## x.12 0.22419 1.29372 0.173 0.863
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 788.6 on 135 degrees of freedom
## (322 observations deleted due to missingness)
## Multiple R-squared: 0.04529, Adjusted R-squared: -0.04664
## F-statistic: 0.4926 on 13 and 135 DF, p-value: 0.9257
##
## AIC and BIC values for the model:
## AIC BIC
## 1 2425.883 2470.942From the above model, we observe the following
Residual standard error = 788.6 on 135 degrees of freedom
The model is not significant as the p_value is greater than 5% level of significance
F-statistic of the model = 0.4926
The multiple and adjusted R_sqrd are 4.529 and -4.664 respectively. However the adjusted R_sqrd is very small
Also, no variables except the intercept in the model are significant as they are greater than 5% level of significance
Residual check and Serial Correlation Check using Breusch-Godfrey
checkresiduals(model1.1asx$model)
##
## Breusch-Godfrey test for serial correlation of order up to 17
##
## data: Residuals
## LM test = 136.44, df = 17, p-value < 2.2e-16From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, p_value of breusch-godfrey test is less that 5% level of significance. This concludes serial correlation in the residuals is highly significant.
Multi-collinearity Check
vif_model1.1asx <- vif(model1.1asx$model)
vif_model1.1asx## x.t x.1 x.2 x.3 x.4 x.5 x.6 x.7
## 53.00292 118.33830 122.33823 125.52489 126.19786 128.35765 133.47622 136.12418
## x.8 x.9 x.10 x.11 x.12
## 139.03106 142.47802 141.04707 138.69245 61.50449vif_model1.1asx > 10## x.t x.1 x.2 x.3 x.4 x.5 x.6 x.7 x.8 x.9 x.10 x.11 x.12
## TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUEAs the Variation inflation factor is greater than 10, we conclude the effect of multi-collinearity is high
Polynomial Distributed Lag Model
We can reduce the effect of multi-collinearity using the polynomial distributed lag model
model2asx <- polyDlm(x = as.vector(asx_split2), y = as.vector(asx_split1),
q = 1, k = 1, show.beta = FALSE)
summary(model2asx)##
## Call:
## "Y ~ (Intercept) + X.t"
##
## Residuals:
## Min 1Q Median 3Q Max
## -1654.13 -706.62 -9.17 561.27 2219.74
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3943.03376 211.54676 18.639 <2e-16 ***
## z.t0 0.41169 1.27421 0.323 0.747
## z.t1 -0.09005 2.53981 -0.035 0.972
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 839.7 on 157 degrees of freedom
## (322 observations deleted due to missingness)
## Multiple R-squared: 0.1097, Adjusted R-squared: 0.09832
## F-statistic: 9.669 on 2 and 157 DF, p-value: 0.0001097From the above model, we observe the following
Residual standard error = 839.7 on 157 degrees of freedom
The model is significant as the p_value is less 5% level of significance
F-statistic of the model = 9.669
The multiple and adjusted R_sqrd are 10.97 and 9.832 respectively. However the adjusted R_sqrd is very small
Also, no variables except the intercept in the model are significant as they are greater than 5% level of significance
A polynomial pattern (q = 1, k = 1) is imposed on the lag distribution i.e. equivalent to represent beta parameters with Kth order polynomial model time.
model2.2asx <- polyDlm(x = as.vector(asx_split2), y = as.vector(asx_split1),q = 4, k = 4, show.beta = FALSE)
summary(model2.2asx)##
## Call:
## "Y ~ (Intercept) + X.t"
##
## Residuals:
## Min 1Q Median 3Q Max
## -1574.26 -685.10 -6.82 528.30 2168.22
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4118.08645 214.92803 19.160 <2e-16 ***
## z.t0 0.03279 1.28010 0.026 0.980
## z.t1 -2.20885 15.15710 -0.146 0.884
## z.t2 3.30652 19.49855 0.170 0.866
## z.t3 -1.44009 7.84269 -0.184 0.855
## z.t4 0.19029 0.97843 0.194 0.846
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 825.6 on 151 degrees of freedom
## (322 observations deleted due to missingness)
## Multiple R-squared: 0.08576, Adjusted R-squared: 0.05548
## F-statistic: 2.833 on 5 and 151 DF, p-value: 0.01784From the above model, we observe the following
Residual standard error = 825.6 on 151 degrees of freedom
The model is significant as the p_value is less 5% level of significance
F-statistic of the model = 2.833
The multiple and adjusted R_sqrd are 8.576 and 5.548 respectively. However the adjusted R_sqrd is very small
Also, no variables except the intercept in the model are significant as they are greater than 5% level of significance
Residual check and Serial Correlation Check using Breusch-Godfrey
checkresiduals(model2.2asx$model)
##
## Breusch-Godfrey test for serial correlation of order up to 10
##
## data: Residuals
## LM test = 145.02, df = 10, p-value < 2.2e-16From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, p_value of breusch-godfrey test is less that 5% level of significance. This concludes serial correlation in the residuals is highly significant.
Multi-collinearity Check
vif_model2.2asx <- vif(model2.2asx$model)
vif_model2.2asx## z.t0 z.t1 z.t2 z.t3 z.t4
## 1454.893 826492.983 12368348.542 22307303.143 4361029.001vif_model2.2asx > 10## z.t0 z.t1 z.t2 z.t3 z.t4
## TRUE TRUE TRUE TRUE TRUEAs the Variation inflation factor is greater than 10, we conclude the effect of multi-collinearity is high
Koyck Distributed Lag Model
model3asx <- koyckDlm(x = as.vector(asx_split1), y = as.vector(gold) + as.vector(oil) + as.vector(copper))
summary(model3asx, diagnostics = TRUE)##
## Call:
## "Y ~ (Intercept) + Y.1 + X.t"
##
## Residuals:
## Min 1Q Median 3Q Max
## -1906.48 -196.75 -31.89 199.81 2961.68
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 141.40687 215.59162 0.656 0.513
## Y.1 0.95823 0.01893 50.607 <2e-16 ***
## X.t 0.04010 0.05285 0.759 0.449
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 477.7 on 157 degrees of freedom
## Multiple R-Squared: 0.9599, Adjusted R-squared: 0.9594
## Wald test: 1880 on 2 and 157 DF, p-value: < 2.2e-16
##
## Diagnostic tests:
## df1 df2 statistic p-value
## Weak instruments 1 157 2019.209407 1.541577e-91
## Wu-Hausman 1 156 6.313068 1.300128e-02
##
## alpha beta phi
## Geometric coefficients: 3385.231 0.04010146 0.9582283From the above model, we observe the following
Residual standard error = 477.7 on 157 degrees of freedom
The model is significant as the p_value is less 5% level of significance
The multiple and adjusted R_sqrd are 95.99 and 95.94 respectively. The adjusted R_sqrd are comparitively greater than the Distributed Lag & Polynomial Distributed Lag model
Also, variable Y.1 in the model is significant to 5% level of significance. The beta and phi values are 0.0401 and 0.9582 respectively
The weak instruments p_value suggests that the model is significant at first stage of least squares fitting
Wu-Hausman test is used to compare the consistency of instrumental variable estimates to ordinary least square estimates. The presence of endogeniety tells us that instrumental variable estimate is consistent and ordinary least square estimate is inconsistent.
The Wu-Hausman test is significant as the p_value is less than 5% level of significance. Hence, we reject the null hypothesis that correlation between explanatory vairable and error term is zero or in other words endogeniety is not present.
Therefore, we conclude that there is significant correlation between explanatory variable and error term is present at 5% level of significance.
Residual Check
checkresiduals(model3asx$model)
From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Multi-collinearity Check
vif_model3asx <- vif(model3asx$model)
vif_model3asx## Y.1 X.t
## 1.443737 1.443737vif_model3asx > 10## Y.1 X.t
## FALSE FALSEThe variation inflation factor is less than 5 when value of q = 2, we conclude the effect of multi-collinearity is not present.
Auto Regressive Distributed Lag Model
model4asx <- ardlDlm(x = as.vector(asx_split2), y = as.vector(asx_split1),
p = 1, q =1)
summary(model4asx)##
## Time series regression with "ts" data:
## Start = 2, End = 483
##
## Call:
## dynlm(formula = as.formula(model.text), data = data, start = 1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2912.60 -96.84 21.21 148.36 791.49
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 242.43332 67.14254 3.611 0.000338 ***
## X.t 0.08502 0.04104 2.072 0.038841 *
## X.1 -0.08243 0.04110 -2.006 0.045446 *
## Y.1 0.94935 0.01388 68.376 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 268.3 on 478 degrees of freedom
## Multiple R-squared: 0.9095, Adjusted R-squared: 0.9089
## F-statistic: 1601 on 3 and 478 DF, p-value: < 2.2e-16From the above model, we observe the following
Residual standard error = 268.3 on 478 degrees of freedom
The model is significant as the p_value is less 5% level of significance
F-statistic of the model = 1601 on 3 and 478 Degrees of freedom
The multiple and adjusted R_sqrd are 90.95 and 90.89 respectively. The adjusted R_sqrd of this model is comparitively higher than DLM, Polynomial DLM and Koyck models.
Also, all the variables in the model are significant as they are smaller than 5% level of significance
Residuals Check
checkresiduals(model4asx$model)
##
## Breusch-Godfrey test for serial correlation of order up to 10
##
## data: Residuals
## LM test = 18.298, df = 10, p-value = 0.05014The breusch-godfrey test has a p_value greater than 5% level of significance. Hence, it states we reject the null hypothesis that variance is not changing in the residuals.
From the time series plot and histogram of the standardized residuals, we observe a non-random pattern and high residual values which violate general assumptions
Also, we can infer that the standardized residuals in the histogram are distributed around zero mean levels
The ACF plot of the residuals do not suggest the existence of Auto-correlation within error terms
Multi-collinearity Check
vif_model4asx <- vif(model4asx$model)
vif_model4asx## X.t L(X.t, 1) L(y.t, 1)
## 91.854920 91.933838 1.024841vif_model4asx > 10## X.t L(X.t, 1) L(y.t, 1)
## TRUE TRUE FALSEThe variation inflation factor is less than 5 for coefficient Y.t when value of q = 1.
Determining the values of p and q using AIC and BIC values
for(i in 1:5){
for(j in 1:5){
model4.1asx <- ardlDlm(x = as.vector(asx_split2),
y = as.vector(asx_split1), p = i, q = j)
cat("p:",i,"q:",j,"AIC:",AIC(model4.1asx$model), "BIC:",
BIC(model4.1asx$model),"\n")
}
}## p: 1 q: 1 AIC: 6764.712 BIC: 6785.602
## p: 1 q: 2 AIC: 6745.169 BIC: 6770.224
## p: 1 q: 3 AIC: 6730.729 BIC: 6759.946
## p: 1 q: 4 AIC: 6719.373 BIC: 6752.746
## p: 1 q: 5 AIC: 6706.985 BIC: 6744.511
## p: 2 q: 1 AIC: 6749.959 BIC: 6775.014
## p: 2 q: 2 AIC: 6744.981 BIC: 6774.212
## p: 2 q: 3 AIC: 6730.858 BIC: 6764.249
## p: 2 q: 4 AIC: 6719.477 BIC: 6757.023
## p: 2 q: 5 AIC: 6707.166 BIC: 6748.862
## p: 3 q: 1 AIC: 6737.899 BIC: 6767.115
## p: 3 q: 2 AIC: 6733.274 BIC: 6766.665
## p: 3 q: 3 AIC: 6732.348 BIC: 6769.912
## p: 3 q: 4 AIC: 6721.016 BIC: 6762.733
## p: 3 q: 5 AIC: 6708.682 BIC: 6754.547
## p: 4 q: 1 AIC: 6726.459 BIC: 6759.832
## p: 4 q: 2 AIC: 6721.97 BIC: 6759.516
## p: 4 q: 3 AIC: 6721.158 BIC: 6762.875
## p: 4 q: 4 AIC: 6722.852 BIC: 6768.741
## p: 4 q: 5 AIC: 6710.564 BIC: 6760.6
## p: 5 q: 1 AIC: 6715.302 BIC: 6752.829
## p: 5 q: 2 AIC: 6710.779 BIC: 6752.475
## p: 5 q: 3 AIC: 6709.946 BIC: 6755.811
## p: 5 q: 4 AIC: 6711.651 BIC: 6761.686
## p: 5 q: 5 AIC: 6712.563 BIC: 6766.768We observe that p = 1 and q = 5 give the smallest AIC and BIC values.
best_model_asx_all <- ardlDlm(x = as.vector(asx_split2), y = as.vector(asx_split1),
p = 5, q = 1)
summary(best_model_asx_all)##
## Time series regression with "ts" data:
## Start = 6, End = 483
##
## Call:
## dynlm(formula = as.formula(model.text), data = data, start = 1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2884.31 -102.27 18.85 149.50 786.82
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 257.202235 69.064158 3.724 0.00022 ***
## X.t 0.067481 0.043097 1.566 0.11807
## X.1 -0.006041 0.068950 -0.088 0.93022
## X.2 -0.024334 0.069549 -0.350 0.72658
## X.3 -0.010001 0.069546 -0.144 0.88572
## X.4 -0.017381 0.068991 -0.252 0.80121
## X.5 -0.008330 0.043211 -0.193 0.84722
## Y.1 0.946784 0.014216 66.602 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 269.1 on 470 degrees of freedom
## Multiple R-squared: 0.9068, Adjusted R-squared: 0.9054
## F-statistic: 653.3 on 7 and 470 DF, p-value: < 2.2e-16From the above model, we observe the following
Residual standard error = 269.1 on 470 degrees of freedom
The model is significant as the p_value is less 5% level of significance
F-statistic of the model = 653.3 on 7 and 470 Degrees of freedom
The multiple and adjusted R_sqrd are 90.68 and 90.54 respectively. The adjusted R_sqrd of this model is comparitively higher.
Variables Y.1 in the model are significant as they are smaller than 5% level of significance
Residuals Check
checkresiduals(best_model_asx_all$model)
##
## Breusch-Godfrey test for serial correlation of order up to 11
##
## data: Residuals
## LM test = 19.577, df = 11, p-value = 0.05148