OVERVIEW
Within the technical analysis family, econometric models are unique because they belong to the only category that generates a continuous stream of discrete numeric values as the forecast. For example, if the analyst has determined that a particular time series exhibits distinctly linear properties, then the following linear regression model should be used:
where
x = the independent variable, time
Y(x) = the dependent variable, the price at time index x
A = the slope
B = the intercept
ε = the error factor whose sum approximates zero
The crux of this regression is based on a fundamental trigonometric identity, specifically the following multiple-angle relationship:
Unfortunately, very few security time series exhibit a distinct single-cycle property for prolonged periods of time. However, the sinusoidal regression may be applied iteratively. That is, calculate the primary cycle coefficients, and remove that cycle from the original time series. Then perform the regression a second or third time.
Autoregression has numerous supporters in the realm of technical analysis. It also has several variations and enhancements, such as the autoregressive integrated moving-average (ARIMA) time-series model introduced by George Box and Gwilym Jenkins in the early 1970s. This model frequently is designated as the ARIMA(p, d, q) model, where p is the autoregressive order, d is differencing order, and q is the moving-average order. Traders interested in more information should refer to Box and Jenkins’ book, Time Series Analysis: Forecasting & Control (Prentice-Hall, 1994). Readers who prefer a less advanced compendium should start with The Analysis of Time Series: An Introduction, by Chris Chatfield (CRC Press, 2003).
In addition, several statistical regression models (such as logistical and exponential) have been performed on securities data, but most return low correlation coefficients except over very short periods of time. See Chatfield’s book (referenced above) for further details.
Within the technical analysis family, econometric models are unique because they belong to the only category that generates a continuous stream of discrete numeric values as the forecast. For example, if the analyst has determined that a particular time series exhibits distinctly linear properties, then the following linear regression model should be used:
where
x = the independent variable, time
Y(x) = the dependent variable, the price at time index x
A = the slope
B = the intercept
ε = the error factor whose sum approximates zero
By solving for the regression coefficients A and B, the trader can
estimate the next value in the time series Y(·) by incrementing the value of x in the linear model.
SIMPLE SINUSOIDAL MODEL
If security prices were not cyclical, they would tend to go off the top or bottom of the charts. This alone justifies the examination of a simple sinusoidal model. The current method identifies the most dominant sinusoidal in the time series using the conventional model:
If security prices were not cyclical, they would tend to go off the top or bottom of the charts. This alone justifies the examination of a simple sinusoidal model. The current method identifies the most dominant sinusoidal in the time series using the conventional model:
where
x = the independent variable, time
Y(x) = the dependent variable, the price at time index x
A = cosine amplitude
B = sine amplitude
Θ = frequency, expressed as cycles per time unit
μ = the arithmetic mean of the time series
x = the independent variable, time
Y(x) = the dependent variable, the price at time index x
A = cosine amplitude
B = sine amplitude
Θ = frequency, expressed as cycles per time unit
μ = the arithmetic mean of the time series
The crux of this regression is based on a fundamental trigonometric identity, specifically the following multiple-angle relationship:
Once the frequency has been isolated and extracted, the two amplitudes can be calculated relatively simply.
Unfortunately, very few security time series exhibit a distinct single-cycle property for prolonged periods of time. However, the sinusoidal regression may be applied iteratively. That is, calculate the primary cycle coefficients, and remove that cycle from the original time series. Then perform the regression a second or third time.
FOURIER TRANSFORM
The fast Fourier transform is another popular method among technical analysts for extracting cycles from a time series. The basic assumption is that any (well-behaved) curve can be approximated as the sum of a finite number of sinusoidals and is based on the following Fourier series:
The fast Fourier transform is another popular method among technical analysts for extracting cycles from a time series. The basic assumption is that any (well-behaved) curve can be approximated as the sum of a finite number of sinusoidals and is based on the following Fourier series:
The transform operations calculate the values for the cosine amplitudes A and the sine amplitudes B in a similar fashion to the simple trigonometric regression above. Most analysts prefer to download an Internet utility to handle the complexities rather than code it themselves. Traders who are interested in more details should refer to Fourier Analysis, by Murray R. Spiegel, in the Schaum Outline Series (1974).
AUTOREGRESSION
The premise behind autoregressive methods is that previous values in the time series directly influence the current value in the time series. Mathematically, this can be expressed as
The premise behind autoregressive methods is that previous values in the time series directly influence the current value in the time series. Mathematically, this can be expressed as
where
x = the time increment
Y(x) = the price at time index x
A = the first regression coefficient
B = the second regression coefficient
C = the third regression coefficient
ε = the error factor, whose sum approximates zero
x = the time increment
Y(x) = the price at time index x
A = the first regression coefficient
B = the second regression coefficient
C = the third regression coefficient
ε = the error factor, whose sum approximates zero
This equation infers that the time-series closing price on any given day is the sum of the closing prices on the three previous days, all adjusted by regression coefficients. The number of independent
variables on the right side of the equation determines the autoregressive order of the model.
variables on the right side of the equation determines the autoregressive order of the model.
Autoregression has numerous supporters in the realm of technical analysis. It also has several variations and enhancements, such as the autoregressive integrated moving-average (ARIMA) time-series model introduced by George Box and Gwilym Jenkins in the early 1970s. This model frequently is designated as the ARIMA(p, d, q) model, where p is the autoregressive order, d is differencing order, and q is the moving-average order. Traders interested in more information should refer to Box and Jenkins’ book, Time Series Analysis: Forecasting & Control (Prentice-Hall, 1994). Readers who prefer a less advanced compendium should start with The Analysis of Time Series: An Introduction, by Chris Chatfield (CRC Press, 2003).
OTHER ECONOMETRIC MODELS
There exist a number of other econometric models that have been applied to financial time series. For example, the Holt-Winters model is a combination of a linear trend model and a seasonal model. A recent addition to time-series analysis is the generalized autoregressive conditional heteroskedacity model (GARCH), which attempts to improve on the ARIMA model by incorporating skew analysis of the data.
There exist a number of other econometric models that have been applied to financial time series. For example, the Holt-Winters model is a combination of a linear trend model and a seasonal model. A recent addition to time-series analysis is the generalized autoregressive conditional heteroskedacity model (GARCH), which attempts to improve on the ARIMA model by incorporating skew analysis of the data.
In addition, several statistical regression models (such as logistical and exponential) have been performed on securities data, but most return low correlation coefficients except over very short periods of time. See Chatfield’s book (referenced above) for further details.
