ECON 20110/30370: Econometrics Semester 2 – Practical Session Worksheet 4

This practical session explores the dynamic relationship between gold and silver prices, focusing on unit root tests, spurious regressions, and Granger causality.

1    Load the Data

We will analyse the ‘GoldSilver’ dataset from the ‘AER ’ library, comprising daily spot prices for gold (per 0.1 oz) and silver (per 10 oz) from December 30, 1977, to December 31, 2012.  This multivariate dataset provides a rich basis for our exploration.

To load this dataset into R, execute the following commands:

library(AER)

data("GoldSilver")

1.1    Data Preparation

The data preparation involves log transforming the series to stabilise variance and compress the scale of the data.  This approach mitigates the disparity between small and large values, ensuring a more meaningful analysis. Furthermore, considering the extensive span of the dataset and to focus our investigation on more recent trends and dynamics between gold and silver prices, we concentrate on a subset of the data post-1990 using the ‘window’ function.

Execute the following R code to prepare the data:

lgs  <-  log(GoldSilver)

lgs78  <-  window(lgs,  start  =  as.Date("1990-01-01"))

gold  <-  lgs78[,1]

silver  <-  lgs78[,2]

The final two lines of code precisely extract the ‘gold’ and ‘silver’ series from the adjusted multivariate dataset ‘lgs78’, ensuring that our analysis is targeted and relevant.

2    Loading the Required Packages

To effectively carry out our analysis, we will use several R packages for dynamic linear modelling and hypothesis testing. Load the necessary packages into your R session as follows:

library(dynlm)

library(car)

This enables the use of ‘dynlm()’ for dynamic modelling and ‘linearHypothesis()’ for performing F-tests, among other analytical capabilities.

3    Analysis of Gold and Silver Prices

This exercise focuses on the dynamic relationship between gold and silver prices, with a focus on unit root tests, the detection of spurious regressions, and the understanding of Granger causality.  Our objective is to interpret the results of statistical tests to gain insights into the properties of the time series data and their relationship.

3.1    Testing for Unit Roots

Determining whether the gold and silver price series are integrated — that is, if they contain unit roots — is our first analytical step. This is crucial for understanding the time series’ characteristics and selecting the appropriate modelling techniques.

1.  Visual Inspection:

Start by closely examining the plot of the log-transformed prices. Observe and discuss any discernible patterns, and compare the two series with respect to their volatility and trends.   This initial visual inspection is an essential preliminary step for gaining insight into the underlying behaviour of the series.

2.  Performing Unit Root Tests:

The Augmented Dickey-Fuller (ADF) test will help us assess the presence of unit roots in both gold and silver price series.

(a)  The use of information criteria such as the  BIC and AIC has determined that an autoregressive model of order one, AR(1), is suitable for both the gold and silver price series.

Discuss the implications of this finding for the number of unit roots present in each series.  (You may assume that the innovation term comes from an i.i.d. white noise process.)

(b)  What should be the dependent variable in the ADF regression for testing unit roots in prices? Based on AR(1) modelling for the level, consider the appropriate number of lags of the first difference that should be included in the ADF regression for each series.

(c)  Investigate the number of stochastic trends in the gold series using a general-to-specific approach: i.  Trend-inclusive ADF Test:

Start with an ADF regression that includes both a trend and a constant:

Δgoldt  = α0 +δ · t + π · goldt − 1 +ut .

A.  Discuss the implications of the test results for the presence of unit roots, focusing on what rejection and failure to reject the null hypothesis imply about the series’ properties.

B.  Use this code to fit the model:

gold_ADFtrend  <-  dynlm(diff(gold)  ~  L(gold,  1)+trend(gold)) summary(gold_ADFtrend)

(Note: The ‘ diff(gold)’function calculates the first difference of the series, while ‘trend(gold) ’ adds a time trend according to the length of the series.)

To test the joint hypothesis that π = δ = 0, execute:

linearHypothesis(gold_ADFtrend,

c("L(gold,  1)  ",  "trend(gold)"))

ii.  Constant-only ADF Test:

Next, examine an ADF model with only a constant:

Δgoldt  = α0 + π · goldt − 1 +ut

using this code:

gold_ADFconstant  <-  dynlm(diff(gold)  ~  L(gold,  1))

summary(gold_ADFconstant)

To test the joint hypothesis that π = α0  = 0, use:

linearHypothesis(gold_ADFconstant,

c("L(gold,  1)  ",  "(Intercept)"))

iii.  No-constant ADF Test:

Lastly, consider an ADF model without a constant or trend:

Δgoldt  = π · goldt − 1 +ut

Implement with:

gold_ADFnon  <-  dynlm(diff(gold)  ~  0+L(gold,  1))

summary(gold_ADFnon)

(Note: Specifying ‘0’ forces the exclusion of an intercept.)

iv.  Based on the 5% significance level, decide on the most appropriate test specification for determining the presence of a unit root in the level of the gold price series’ .

v.  Determine the necessity of testing for unit roots in the first differences.  What do the ADF test resultsimply about the gold series’ integration order?

(d)  Repeat the analysis for the silver series, applying the general-to-specific approach to conclude its order of integration.

3.2    Detecting Spurious Regression

A common challenge in time series analysis is distinguishing between spurious relationships and genuine correlations, especially with non-stationary data.  Spurious regression can lead to misleading conclusions about the economic relationships between variables.

1.  Naive Regression Analysis: