时间序列分析

Gold prices are a unit-root based on the Dickey-Fuller test indicated by a p-value of 0.999. What should we do now? This also means that we have a stochastic trend. How do we proceed if the time series has a unit-root? One possibility is to first-difference the time series; hence, we deduct the lagged dependent variable to create a time series in first-differences that reflect the change in levels. 如果时间序列有单位根，可能的解决方法就是对时间序列求一阶差分。
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Lecture 5 5. Time series analysis: Univariate models
5.1 Learning outcomes
Deterministic trends Classical time series model Stationarity Testing for unit roots ARIMA specification ARCH / GARCH models Application: electricity consumption in the UK
3 Dr Gerhard Kling, Quantitative Research Methods in Finance, University of Southampton
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(4) ������������ = ������������−1 + ������������ ⟹ ∆������������ = ������������ Let’s take the first-difference of the gold price, and re-run the Dickey-Fuller test based on the first-difference. Figure 3 shows the first-differenced time series.
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100 gold
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Dr Gerhard Kling, Quantitative Research Methods in Finance, University of Southampton
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5.3 Stationarity and testing for unit roots 平稳性和单位根检验
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100 gold
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The first figure seems to suggest a good model fit – but even forecasting for short time horizons is very unreliable. For instance, Figure 2 shows the forecast for the next 10 months. Figure 2: Forecasting the next 10 months
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Testing for a unit-root implies that the coefficient γ is equal to zero. The Dickey-Fuller test provides critical values for the associated t-statistic. Model 3 is a regression model with lagged dependent variable and hence OLS won’t produce reliable test statics, as exogeneity is violated. WARNING! The Dickey-Fuller tests uses the null hypothesis that the time series is a unit-root; hence, if you cannot reject the null hypothesis you cannot reject that the time series is a unitroot.
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Dr Gerhard Kling, Quantitative Research Methods in Finance, University of Southampton
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Figure 1: Deterministic model for monthly gold prices
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5.2 The classic time series model
The classic time series model assumes that a time series which refers to a sequence of realisations {yt} can be decomposed into a trend, a cyclical component and a random noise. (1) ������������ = ������������������������������ + ������������������������������ + ������������������������������ Traditionally, the trend component has been estimated using a linear time trend or higher order polynomials of time. As we only use time to explain the sequence, we call this approach a univariate approach. Accordingly, the assumption is that the trend is deterministic and depends on the time dimension. Somehow this approach was not too successful in forecasting time series. Forecasting is the main purpose of time series analysis, and hence a method that does not do a good job in forecasting is not of great use. 趋势部分用线性时间趋势或者时间的高阶多项式来估计。由于只用时间来解释序列， 该方法为单变量方法。 To illustrate the problems with deterministic trends, one needs to consider the mathematical behaviour of a deterministic time series (or better an iteration sequence). The main issue is the out-of-sample behaviour of deterministic trend models. I used monthly gold price data from 1978 to 2011 and constructed a deterministic trend model based on the first 400 observations. I then tried to forecast the remaining 106 observations. Figure 1 shows that the model looks very reasonable within the sample and achieved an R-squared of 80%. However, when I tried to forecast for the next years, the forecasts were quite useless as shown in Figure 2. The problem is the long-term behaviour of the deterministic iteration sequence. 问题在于 确定性迭代序列的长期行为 R-squared 拟合度 Fitted values 拟合值
The concept of stationarity contends that a time series is said to be stationary if and only if it has a constant mean, variance and covariance structure. Put differently, the realisations of the sequence {yt} are drawn from the same random variable. In time series econometrics we call this a data generating process. Hence, when we observe 20 realisations we are able to estimate the moments (mean and variance) and the covariance structure (relationship between realisations over time) is constant. Stationarity is a prerequisite for many analyses, as a nonstationary time series does not have defined moments. Without defined moments, many analyses are rendered useless. 当且仅当有恒定的均值，方差和协方差时，该时间序列才是平稳的。 平稳性分析是做分析的先决条件。 There are different methods to test for stationarity – so-called unit root tests. What is a unitroot? The following equation is a unit-root (or random walk) if the coefficient a=1. 单位根（随机游走）检验 (2) ������������ = ������������������−1 + ������������ The error term in equation 2 is a white-noise process, which can have a constant mean (drift component). The error term has a constant variance and is not serially correlated. Actually, it is iid distributed. We can modify equation (2) by taking – yt-1 from both sides following Dickey and Fuller (1979). (3) ∆������������ = ������������ − ������������−1 = (������ − 1) ������������−1 + ������������