W3 Time Series - Lecture notes 3 PDF

Preview

Summary

quick notes on time series...

Description

Step 1: Define the problem - What is the objective? - Who is going to use the data? - What time period? - How accurate should the forecast be? - What form is the forecast? (mean, confidence interval) - How often will forecast be needed? Step 2: gather data - Numerical data - Collective judgement (qualitative data) Step 3: Preliminary analysis Step 4: Choose methods - Principle of parsimony: choose the simplest model possible that still accomplishes the task Step 5: Use and evaluate model - Implement forecast monitoring Time Series Data - Collected over successive increments of time, e.g. every month, week - Data is assumed to NOT be independent, because the previous period affects the next period, which helps us predict the future trend - Data must be in order 1. Stationary time series ● constant mean and constant variance ● Observations are independent of time and of each other ● Special type of stationary time series is a ‘white noise’ -> mean around 0, observations are independent of each other ➢ We want errors to have white noise distribution

2. Trend ● ● ● ●

Growth or decline over time Long term movements in the data series Can be described by straight lines or smooth curves Time series is non-stationary

3. Cycle ● Wavelike fluctuations around trend, not regular and not of a fixed period ● Usually influenced by changes in economic conditions (business cycles)

4. Seasonal ● Pattern that regularly repeats itself, it has a fixed period ● Data influenced by seasonal factors

5. Random Assumption - the underlying patterns / relationships found in the data will not change during the forecasting period Time Series Decomposition The main purpose of using a classical decomposition is to pull apart a time series in order to get a better feel for the patterns within, namely trend and seasonality. However, you can also use the results from the decomposition analysis to forecast future values of the time series. The forecasts from a classical decomposition tend not to be that great, so the technique is often not used for that purpose these days. However, it is still useful to see how it's done, especially since more advanced decomposition techniques such as X-12 ARIMA (not covered on this course) are used for forecasting, and because it's important to understand the principles of forecasting future values.

Y=f(S+T+I) S = seasonal T = trend-cycle I = irregular There are two types of models 1. Additive model, Y = S + T + I - Used when variance is constant - (Under this model, the components have the same units as the original series, and the S and I components are distributed around 0.) 2. Multiplicative model, Y = S × T × I - Used when there is increasing variance - (Under this model, the trend has the same units as the original series, but the seasonal and irregular components are unitless factors, distributed around 1.)

(

Step 0: Decide if you want to use additive or multiplicative model Step 1: Estimate the trend cycle: - Moving average approach Step 2: De-trend the series: - Additive Model: Y-T= S+I - Multiplicative Model: Y/T= SxI Step 3: Calculate the unadjusted seasonality Step 4: Calculate the adjusted seasonality Step 5: De-seasonalise the series: - Additive Model: Y-S= T+I - Multiplicative Model: Y/S= TxI Step 6: Use regression to forecast ) Step 7: Analyse the Irregularity: - Additive Model: Y-S-T= I - Multiplicative Model: Y/SxT= I

STEP 1: Estimating T Can either use 1. Moving averages - k is the order (or length) of the moving average -

When k is odd Moving average of order k used to smooth data is denoted by “kMA” If there is a seasonal pattern, choose k to be equal to the seasonal period. This is so that you don’t pick up the seasonal pattern, i.e. smoothing is done Calculations:

➢ The length on either side is length is 1 on either side

-

When k is even Denoted 2×k-MA Calculations:

k−1 , so e.g. if calculating a 3-MA, 2

➢ The length on either side is

k , so e.g. if calculating a 2×12-MA, 2

length is 6 on either side ➢ To calculate the middle value, you need to take _ + 2(_ _ _ _ _+ _ _ _ _ _) + _ and then divide by 2×k

2. Regression methods - More specific because trend is assumed to follow that one equation

Additional note: Why do S and T either centre around 0 or 1? - If an observation sat exactly on the estimated trend line of the time series, then Y = T at that point. If we are carrying out a multiplicative decomposition and we therefore de-trend the series by calculating Y/T we would get the value one at that point. If the observation value at that point is above the estimated trend line, then Y > T at that point and therefore when de-trending the series we would get Y/T > 1. If the observation value at that point is below the estimated trend line, then Y < T at that point and therefore when de-trending the series we would get Y/T < 1. And so we can see that if we have estimated the trend component well, the de-trended values should fluctuate around the value one. -

-

Likewise, if we are carrying out an additive decomposition: If an observation sat exactly on the estimated trend line of the time series, then Y = T at that point. If we de-trend the series by calculating Y - T we would get the value zero. If the observation value at that point is above the estimated trend line, then Y > T at that point and therefore when de-trending the series we would get Y - T > 0. If the observation value at that point is below the estimated trend line, then Y < T at that point and therefore when de-trending the series we would get Y - T < 0. And so we can see that if we have estimated the trend component well, the de-trended values should fluctuate around the value zero...