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Questions and Answers

What does it mean if a time series is integrated of order d, denoted as yt ∼ I(d)?

The series oscillates around zero without a trend.

The series must be differenced d times to achieve stationarity. (correct)

The series is stationary and does not require differencing.

The series has a constant mean and variance.

Which of the following statements accurately describes an I(1) series?

It contains one unit root and requires one differencing to become stationary. (correct)

It constantly crosses its mean value.

It has two unit roots and is also stationary.

It is stationary and requires no differencing.

What is the primary characteristic of I(0) series?

They require multiple differencing to achieve stationarity.

They contain two unit roots and do not exhibit mean reversion.

They frequently cross their mean value. (correct)

They wander far from their mean without crossing it.

For the series defined as yt = φyt-1 + ut, what is the null hypothesis when testing for a unit root?

φ = 1 Signup and view all the answers

Which of the following statements is true regarding the behavior of I(1) and I(2) series?

I(1) and I(2) series can wander far from their mean values. Signup and view all the answers

What phenomenon can lead to infinite persistence of shocks?

Non-stationary series Signup and view all the answers

Why is testing for non-stationarity important in regression analysis?

It prevents spurious regressions among unrelated variables. Signup and view all the answers

Which model is commonly used to characterize non-stationarity?

Random walk model with drift Signup and view all the answers

What happens to the standard assumptions for asymptotic analysis when regression variables are non-stationary?

They are violated, affecting validity of hypothesis tests. Signup and view all the answers

In the context of non-stationarity, what does the term 'explosive process' refer to?

A process characterized by a growth rate greater than 1 Signup and view all the answers

Which assumption is invalidated if the variables in a regression model are non-stationary?

The t-ratios will follow a t-distribution. Signup and view all the answers

Which condition does NOT contribute to spurious regression results?

Randomization of the dataset Signup and view all the answers

What is covariance stationarity primarily concerned with?

The consistency of variance and autocovariance over time Signup and view all the answers

What effect does a value of $eta$ greater than 1 have on shocks in a system?

Shocks become more influential and persistent. Signup and view all the answers

In the AR(1) model without drift, how do successive substitutions affect the relationship of $y_t$ with past values?

$y_t$ is expressed as a function of past values and multiple shocks. Signup and view all the answers

What is the treatment required for deterministic non-stationarity to achieve stationarity?

Detrending the series. Signup and view all the answers

How can stationarity be induced in the first case of stochastic non-stationarity?

By differencing the series once. Signup and view all the answers

What is a characteristic property of an autoregressive (AR) model when $eta > 1$?

Shocks compel the series farther away from the mean over time. Signup and view all the answers

What mathematical operation can express the differenced series in the context of stochastic non-stationarity?

$ abla y_t = eta + u_t$ Signup and view all the answers

In the representation of $y_t$, what does the term $eta t$ indicate in deterministic non-stationarity?

It captures trends over time. Signup and view all the answers

What type of series requires different treatments when inducing stationarity?

Both stochastic and deterministic non-stationary series. Signup and view all the answers

What is the main consequence of differencing a trend-stationary series?

It introduces an MA(1) structure into the errors. Signup and view all the answers

If a series exhibits stochastic non-stationarity, what would happen if you attempt to detrend it incorrectly?

You will fail to remove the non-stationarity. Signup and view all the answers

Which of the following processes describes a random walk?

It is a non-deterministic trend. Signup and view all the answers

What does the term 'unit root' refer to in the context of non-stationary time series?

A statistical characteristic indicating integration of order one. Signup and view all the answers

Which autoregressive process result shows a value of $eta=1$?

The series exhibits explosive behavior. Signup and view all the answers

In a deterministic trend process, how does the series behave over time?

It follows a predictable linear pattern. Signup and view all the answers

How does a white noise process appear on a plot?

It shows spikes around a constant mean. Signup and view all the answers

What is the primary focus of the stochastic non-stationarity model discussed?

Understanding economic and financial series behavior. Signup and view all the answers

What happens when you try to detrend a series that includes multiple unit roots?

Detrending has no effect on non-stationarity. Signup and view all the answers

Which autocorrelation value would indicate a stationary process?

It decreases rapidly toward zero. Signup and view all the answers

Study Notes

Modeling Long-Run Relationships in Finance

Modeling long-run relationships in finance is essential for understanding the behavior and properties of financial variables. Stationarity of a series significantly influences its behavior and properties. For instance, persistence of shocks in non-stationary series is infinite.
Spurious regressions can occur when two variables trend over time. This can lead to a high R² in their regression even if there is no actual relationship.
Non-stationary variables in regression models invalidate the standard assumptions for asymptotic analysis typically used in regression analysis. Consequently, standard t-ratios don't follow a t-distribution and hypothesis tests about regression parameters become invalid.
A graph illustrating the value of R² shows a substantial spread across various datasets, emphasizing that spurious regressions are inherently unreliable.
A graph demonstrating t-ratios reveals another issue within the regression of non-stationary data, further highlighting the unreliability in analysis of non-stationary data.

Stationarity & Unit Root Testing

Stationarity is crucial for reliable analysis; if a series isn't stationary, typical assumptions, and therefore typical conclusions from, regression tests become inaccurate.
Non-stationary series can exhibit infinite shock persistence.
Spurious regressions result when regressing one trending variable on another, even if no real relationship exists.
Non-stationary variables in regression models violate the standard assumptions used in asymptotic analysis. The resulting t-ratios do not follow a t-distribution's properties; thus, regression parameter hypothesis tests are unreliable.

Two types of Non-Stationarity

There are multiple definitions of non-stationarity.
The context is focused on weak forms or covariance stationarity.
Two models frequently characterize non-stationarity:
Random walk model with drift: Yt = μ + y t - 1 + u t
Deterministic trend process: Yt = a + βt + u t
u, is independently and identically distributed (iid) in both cases.

Stochastic Non-Stationarity

The model (1) can be expanded to include an explosive process: Yt = μ + φyt-1 + u t where |φ| > 1 .
The explosive case is usually disregarded because it doesn't reflect many financial data characteristics.
Shocks in an explosive process become increasingly large with time, impacting results in a complex manner. This is not the usual way that financial time series behave.

Stochastic Non-stationarity: The Impact of Shocks

AR(1) with no drift is considered: Yt = φyt-1 + u t
Substituting back, we show how past shocks influence present values over a sequence of periods from the first period in time.

The Impact of Shocks for Stationary and Non-Stationary Series

Three cases are considered.
- φ < 1: shocks diminish over time.
- φ = 1: shocks persist indefinitely.
- φ > 1: shocks gain in strength with time.

Detrending a Stochastically Non-stationary Series

To induce stationarity, a stochastically non-stationary series requires differencing.
The trend-stationary model requires detrending.
Differencing a stochastically nonstationary series removes the non-stationarity, but it introduces an MA(1) structure into the error terms, altering the structure of the regression to consider for the analysis.
Non-stationary series can include both types of non-stationarity. Differencing removes stochastic non-stationarity, but this can change the properties of the regression, while detrending removes deterministic non-stationarity.

Detrending A Series: Using The Right Method

Differencing trend-stationary series induces a stationary series but introduces a MA(1) model into the errors. Detrending a stochastically non-stationary series does not produce stationarity. This is why it is important to consider the type of non-stationarity in your data.

Sample Plots for Various Stochastic Processes

Visual representations of different stochastic processes (including white noise, random walks, and a deterministic trend process) provided to understand the behavior and characteristics exhibited by each in graphs.

Autoregressive Processes with Differing Values of φ

Graphs illustrating AR(1) processes with different values of φ (0, 0.8, 1). Varying φ illustrates the impact on the autoregressive property.

Definition of Non-Stationarity

A non-stationary series requires differencing d times to become stationary. This is written as yt~I(d). I(0) is stationary. I(1) has one unit root. An I(2) series contains 2 unit roots.

Characteristics of I(0), I(1), and I(2) Series

I(2) series require differencing twice to establish stationarity.
I(1) and I(2) series have limited mean-crossing, and their mean values fluctuate significantly through time.
I(0) series frequently cross their mean value.
Many economic/financial series exhibit single unit roots.

How do we test for a unit root?

The Dickey-Fuller (DF) test is a common method for testing a unit root on time series data.
A fundamental goal of the DF test is to empirically determine whether a series contains a unit root or not. A unit root implies a characteristic that the series' evolution through time or over some periods or epochs has not converged to some constant mean value.
A related concept and extension for checking for trends exists using τ, τμ, and τ₀ (tau) tests.

Different Forms for the DF Test Regressions

Different models, including those accounting for drift and time trends, are used during the evaluation when employing the Dickey-Fuller test.

Computing the DF Test Statistic

The DF test statistic does not follow the standard t-distribution with the null hypothesis being that of non-stationarity. As such, critical values are obtained using Monte Carlo experiments on various situations that exhibit non-stationarity for different significance levels.

Critical Values for the DF Test

Tables showing critical values for the DF test at different significance levels for various types of regressions and how they are applied when assessing the statistical significance of the results when the null hypothesis is that of non-stationarity.

The Augmented Dickey-Fuller (ADF) Test

The ADF is an expansion on the DF test, used to account for possible autocorrelation in the error terms, making it useful for data with autocorrelation. The ADF test introduces lags in the independent variables.

Testing for Higher Orders of Integration

A DF test is useful to rule out that a time series may be stationary, i.e., may not contain a unit root.
This allows for a more complete assessment of the potential order (2) of integration through the test (A²yₜ on ▲y t-1).

The Phillips-Perron Test

A Phillips-Perron test complements ADF tests, addressing certain issues like autocorrelated residuals in an automatic way.

Criticism of Dickey-Fuller and Phillips-Perron-type Tests

Power of DF/PP tests is weak when the series exhibits stationarity but is close to non-stationarity.
A possible solution is to use both a stationarity test and a unit root test to improve the accuracy of the results.

Stationarity Tests

Stationarity tests typically determine if a given time series is stationary or not based on the null hypothesis that the series is stationary vs the alternative that it is not.
A KPSS test (Kwaitowski, Phillips, Schmidt & Shin, 1992) is an example of a well-used stationarity test.
Comparing results from the ADF/PP with results from the KPSS can confirm findings.

Stationarity Tests(cont'd)

Compare ADF/PP and KPSS results for consistency.
Four possible outcomes in stationarity tests are given: Reject Ho & Do not Reject Ho; Do not Reject Ho & Reject Ho; Reject Ho & Reject Ho; Do not Reject Ho & Do not Reject Ho.

Unit Root Tests with Structural Breaks

Standard DF-type tests lack power in situations with structural breaks, often leading to inaccurate outcomes. The presence of structural breaks can bias the slope parameter estimates toward unity (1). The power to detect a unit root decreases with the degree of the structural break and the size of the sample.

The Perron (1989) Procedure - Background

Perron's approach introduces three different models for specific types of structural breaks to improve the quality of the results during unit root tests.
These models account for:
- Structural break in the level (intercept)
- Structural break in the trend (slope)
- Structural breaks in both the level and trend

The Perron (1989) Procedure - Details

The most general version of Perron's test equation is provided, allowing for breaks in both the level and the trend.
Models are specified for different structural break types to account for potential disruptions during testing.
There are limitations to this approach due to the prerequisite for prior knowledge of the break date. The break date isn't always known in advance, so this can affect the accuracy or validity of the results. This is handled by the more complex approaches that are discussed subsequently.

The Banerjee et al. (1992) and Zivot and Andrews (1992) Procedures - Background

Researchers often use the break date based on data examination, not prior information. The use of data for break date estimation raises concerns about asymptotic theory, prompting new approaches for evaluating cointegration or non-stationarity.

The Banerjee et al. (1992) and Zivot and Andrews (1992) Procedures - Details

Techniques like recursive and rolling regressions are proposed to handle the endogenous determination of break dates in cointegration or non-stationarity testing. This allows the analysis to account for structural breaks. Specific models were developed for different break date estimations.

Further Extensions

Subsequent extensions to the Perron technique allow for multiple structural breaks and adaptive estimation procedures, improving the flexibility and reliability of results that account for structural changes. This type of testing is very similar to Zivot and Andrews' models but more flexible due to allowing breaks under both null and alternative hypotheses.

Testing for Unit Roots with Structural Breaks: Example - EuroSterling Interest Rates

Example case study on testing for unit roots in Euro Sterling interest rates, with possible structural breaks (due to possible changes in monetary policy or exchange rate control changes), and accounting for the impact of those breaks on estimates of the relationship.
Different approaches that allow for breaks (recursive, sequential, rolling) are compared and analyzed to observe significant differences.

Testing for Unit Roots with Structural Breaks in EuroSterling Interest Rates - Results

Different methods are employed. This illustrates a statistical comparison of different approaches or methodologies through tables.

Testing for Unit Roots with Structural Breaks in EuroSterling Interest Rates - Conclusions

Statistical inferences from recursive methods do not reject the null hypothesis at the 10% level.
For the sequential tests, the outcomes are varying, with some cases not rejecting the null (with the trend), but rejecting the null for some cases with the short-term rates, meaning there may be breaks in their mean levels.

Seasonal Unit Roots

Seasonal unit roots are addressed by seasonal differencing (I(d,D)) to achieve stationarity
Osborn's approach is relevant, with a Dickey-Fuller-like seasonal unit root test.
While seasonal patterns exist in macroeconomic series, seasonal unit roots are relatively rare, making dummy variables a more common way for representing these patterns in analysis.

Cointegration: An Introduction

Combining two I(1) variables often produces another I(1) variable.
Combining variables with varying orders of integration yields a variable of the highest order.
The calculation illustrates how variables can be combined in order to analyze relationships.

Linear Combinations of Non-stationary Variables

A linear combination of non-stationary variables often results in undesirable properties, mainly due to non-stationarity and autocorrelation, which hinder accurate analyses.

Definition of Cointegration

Components of a vector of variables become cointegrated if a linear combination of the variables converges to a stationary process (I(d-b)). This means that variables are often non-stationary together but they move in predictable, consistent ways over time.
Often, cointegration is thought of as variables moving together in a predictable way over time

Cointegration and Equilibrium

Examples in finance include spot and futures prices, ratios of relative prices and exchange rates, and equity prices and dividends.
Cointegration reflects equilibrium relationships; the absence of cointegration implies that variables can diverge indefinitely, with no stable long-run relationship.

Equilibrium Correction or Error Correction Models

The common method to deal with non-stationary variables was initially to take the first difference of each variable before any analysis of relationships.
This method, while appearing logical or standard at the time it was suggested by proponents of the approach, has been shown to be problematic due to long-run issues.
Defining the long run in terms of a period where variables are not changing at all (Δyt=0) allows for inclusion of previous value information within the model of the dependent variable.

Specifying an ECM

A practical and relevant method for determining a long-run solution where the previous period's information is used for a given period or epoch by incorporating the difference of the variables into the error correction model, e.g., ∆ yt = β₁∆xt + β₂ ( yt−1 − γxt−1) + ut. This method works to incorporate all required non-stationary information within the model without significantly compromising the quality or the relevance of any results.

Testing for Cointegration in Regression

The regression model for the cointegrating relationship extends to cover multiple variables, with residuals showing if the series is non-stationary versus stationary.
The DF/ADF test can be used on the model's residuals, but critical values are adjusted to account for the model's characteristics when taking into empirical account the effect of the model being non-stationary or stationary.

Testing for Cointegration in Regression: Conclusions

Engle-Granger's test uses different critical values relevant to this approach. Methods such as the Durbin Watson test and the Phillips-Perron approach are also relevant when considering how cointegration can be shown between non-stationary variables.

Methods of Parameter Estimation in Cointegrated Systems: The Engle-Granger Approach

Three primary methods to evaluate cointegration relationships (Engle Granger, Engle Yoo, Johansen).
Engle-Granger 2-step method steps:
Ensure all variables are I(1).
Estimate cointegrating regression via OLS.
Save the residuals from the cointegrating regression.
Test the residuals to establish if they are I(0).
Employ the residuals within an error correction model.

An Example of a Model for Non-stationary Variables: Lead-Lag Relationships Between Spot and Futures Prices

Spot and futures prices often have a correlated relationship, reflecting predictable patterns that extend over the long horizon.

Futures & Spot Data

Case studies provided to apply methodologies described. Data sets for analysis were employed as needed during studies and examples.

Methodology

Theoretical equation provided to define the relationship between the spot and the future price.

Dickey-Fuller Tests on Log-Prices and Returns for High Frequency FTSE Data

Dickey-Fuller statistical results demonstrated on spot and futures markets for the FTSE index.
The output from these tests is directly presented within the relevant sections.

Cointegration Test Regression and Test on Residuals

Cointegration regression model and approach used to determine the cointegrating relationship between spot and futures data values.

Estimated Equation and Test for Cointegration for High Frequency FTSE Data

Results presented in a formatted manner. The results are directly presented in tables, suitable for immediate observation of statistical significance, parameter estimations and evaluation of different findings..

Conclusions from Unit Root and Cointegration Tests

Log F and log S are generally not stationary.
A cointegrating relationship between log Fₜ and log Sₜ may exist. A final stage of the Engle Granger 2-step method is introduced and the overall cointegrating model is formally defined.

Estimated Error Correction Model for High Frequency FTSE Data

Results regarding cointegrating model coefficients and related t-ratios.
Table illustrating coefficients and t-ratios is provided.

Forecasting High Frequency FTSE Returns

Error correction model performance compared with alternative models like ARIMA and VAR in terms of RMSE, MAE, and percent correct forecasts. Table showing comparative performance of different forecasting methods is provided

Can Profitable Trading Rules be Derived from the ECM-COC Forecasts?

Trading strategy involving analyses of predictions for spot returns. This assumes that the original investment remains a constant amount that does not change in any manner, and where the holding in the stock index is zero, and investment earns the risk-free rate.

Spot Trading Strategy Results for Error Correction Model Incorporating the Cost of Carry

Results from the developed trading strategies are presented. Specifically, a table is given indicating results with and without slippage when considering different trading strategies.

Conclusions

The futures prices "lead" the spot market due to index components trading in different frequencies and costs associated with spot market transactions.
The findings demonstrate a long-run relationship between spot and futures prices, but no evidence of arbitrage opportunities.

The Engle-Granger Approach: Some Drawbacks

Unit root and cointegration tests can have low power in finite samples.
Asymmetric treatment of variables.
Difficulties with hypothesis testing for the cointegrating relationship.
The Engle-Granger method has limitations addressed by the Johansen technique.

The Engle & Yoo 3-Step Method

The Engle & Yoo method provides a solution to the EG method's shortcomings. The method adds a third step providing updated estimates of the cointegrating vector and its standard errors to determine better estimates (and more reliable/robust cointegrating vectors) via the iterative process.

The Engle & Yoo 3-Step Method (cont'd)

Handling multiple cointegration cases (many variables potentially linked) is crucial. The Johansen technique is a suitable method for addressing and handling this aspect.

Testing for and Estimating Cointegrating Systems Using the Johansen Technique Based on VARS

The Johansen technique converts a vector autoregression (VAR) to a vector error correction model (VECM) to study cointegrating relationships.

Review of Matrix Algebra necessary for the Johansen Test

Matrix algebra concepts (matrices, characteristic roots/eigenvalues, rank of a matrix) are reviewed, key for understanding the Johansen test.

Review of Matrix Algebra (cont'd)

Matrix algebra concepts (matrices, characteristic roots/eigenvalues, rank of a matrix), and their application to the Johansen test are reviewed.

The Johansen Test and Eigenvalues

Properties of eigenvalues (trace, determinant, rank) of a matrix are reviewed and given in relation to the VAR context.

The Johansen Test and Eigenvalues (cont'd)

Eigenvalues and their interpretation in the context of the Johansen test are examined.

The Johansen Test Statistics

Specific formulas for trace and maximum eigenvalue tests for cointegration rank (r) with relation to eigenvalues (λ).

Decomposition of the Π Matrix

The matrix П's decomposition into cointegrating vectors (contained in matrix β) and loadings (in matrix α) shows the relations between the vectors.

Johansen Critical Values

Critical values for Johansen's statistics depend on the number of non-stationary components and the presence/absence of a constant and/or trend in the models.

The Johansen Testing Sequence

A step-by-step approach is presented when employing the Johansen methodology. This procedure or analysis approach involves iterative tests for increasing r values until a given value of r is not rejected by the test methodology.

Interpretation of Johansen Test Results

Rank (r) of the Π matrix determines whether cointegration or no cointegration is implied or shown.

Hypothesis Testing Using Johansen

Johansen's approach allows for hypothesis testing directly on cointegration vectors (linear combinations).
This allows the testing on cointegrating vectors themselves, unlike the EG approach, in this manner.

Hypothesis Testing Using Johansen (cont'd)

Hypothesis testing becomes more complex with additional considerations and potentially more numerous restrictions in cointegrating relationships.

Cointegration Tests using Johansen: Three Examples

Example evaluating the Purchasing Power Parity (PPP) hypothesis using Johansen's method and data.
Example of the application of Johansen's method. Specific examples to evaluate and illustrate the method.

Example 2: Purchasing Power Parity (PPP)

Example illustrating applications of the Johansen method within a PPP framework to understand equilibrium and cointegration in the context of exchange rates.

Cointegration Tests of PPP with European Data

Table with cointegration test results between currencies using Johansen's method that include the test statistics as well as related critical values for different orders of cointegration.

Example 3: Are International Bond Markets Cointegrated?

Example applying Johansen's methods using bond yield data to illustrate how to cointegrate different markets or how potentially different financial markets or asset classes perform together across time.

Testing for Cointegration Between the Yields

Tests using Johansen procedure's trace test statistics to measure cointegration for international bond yields. A table summarizing the Johansen procedure's outcomes is given.

Testing for Cointegration Between the Yields (cont'd)

Conclusion stating that the Johansen test does not show any cointegrating vectors in the bond yields.

Variance Decompositions for VAR of International Bond Yields

Table presenting variance decompositions from the VAR model for international bond yields across 1-, 5-, 10-, and 20-day time horizons. This provides the breakdown of the contribution from each variable as compared to how those contributions change with time and how variance is explained across markets.

Impulse Responses for VAR of International Bond Yields

Detailed tables illustrate impulse responses of one bond market yield to shocks in other bond market yields in the international bond market example in the time series context. The graphs illustrate the impact of one yield on the other across time.

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This quiz tests your understanding of integrated time series, specifically focusing on the characteristics of I(d), I(1), and I(0) series. You'll be asked about unit roots, null hypotheses, and the behavior of different integrations in time series analysis. Prepare to showcase your knowledge in econometrics!

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