Francis X. Diebold, Elements of Forecasting, 4th Edition

Transcription

1 P1.T2. Quantitative Analysis Francis X. Diebold, Elements of Forecasting, 4th Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM

2 Diebold, Chapter 5 Modeling and Forecasting Trend DESCRIBE LINEAR AND NONLINEAR TRENDS

3 Diebold, Chapter 5 Modeling and Forecasting Trend Describe linear and nonlinear trends. Describe trend models to estimate and forecast trends Compare and evaluate model selection criteria, including mean squared error (MSE), s2, the Akaike information criterion (AIC), and the Schwarz information criterion (SIC). Explain the necessary conditions for a model selection criterion to demonstrate consistency Describe linear and nonlinear trends. Trend is the general direction of movement in any variable which evolves slowly over a period of time. Trends are exhibited in various fields which can be verified empirically, i.e., upon observation. Trends that can be modeled and forecasted are called deterministic, that is, the trend progresses in a such a way that it is highly predictable. A linear trend is one in which the progression of the variable is approximately linear, that is, it changes at a constant rate as in a straight line. A linear trend may either be increasing or decreasing. To describe this trend, consider a simple linear function of time, in which the trend moves in a straight line, i.e., in a linear fashion as the time period increases. The value of the trend at time period t (T t) can be represented by the equation for a straight line (of the form y = c + mx) as: = + where TIME t = t = (1, 2, 3,..., n-1, n) for a sample of size n. The variable TIME is called a time trend or "time dummy and the series is constructed artificially. is called the intercept and it is value of the trend at time t = 0. is the slope of the line, which is positive (negative) if the trend is increasing (decreasing). If the absolute value of is large, the trend s slope is considered to be steep. In the example to the right, the increasing trend has an intercept of = -50 and a slope of =.8, but the decreasing trend has an intercept of =10 and a slope of =

4 Nonlinear (or curved) trends are those in which the variable changes at an increasing or decreasing rate rather than at a constant rate like in linear trends. Nonlinear trends can be expressed in the form of quadratic functions of time (as against linear functions in linear trends) as: = + + Note the linear trend is a special case where = 0 In the figure to the right, we observe the nonlinear trend in monthly volume of shares traded on the New York Stock Exchange (NYSE), in which the volume increases at an increasing rate. Quadratic trends are non-linear By employing a second power of the time variable, we can approximate a certain type of nonlinear trend. These nonlinear quadratic trends can take different forms as illustrated below. These include nonlinearly increasing, nonlinearly decreasing, U shape, and inverted U shape. 4

5 Sometimes a nonlinear trend can be represented in alternate forms. For example, in the NYSE volume series, while the trend of the volume is actually nonlinear, the trend of the logarithm of volume is considered approximately linear as depicted below. Log volume on the New York Stock Exchange When the variable s actual trend is nonlinear, but its logarithmic form is linear, it is called an exponential trend, or log linear trend. Consider the below trend which is nonlinear as described exponentially by its constant growth rate. (Note the difference between the constant rate in linear trends and constant growth rate in exponential trends.) = When the same is transformed in to a logarithmic form, we get a linear function = ( ) + Like quadratic trends, based on the parameter values, exponential trends can vary like increasing or decreasing at an increasing or decreasing rate. While nonlinear trends in some series are well explained by quadratic trend, sometimes, trends in other series are better explained by exponential trend. 5