Week 3 September 15-19

Five Mini-Lectures QMM 510Fall 2014

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Time-Series Analysis ML 3.1

Chapter Contents14.1 Time-Series Components

14.2 Trend Forecasting

14.3 Assessing Fit

14.4 Moving Averages

14.5 Exponential Smoothing

14.6 Seasonality

14.7 Index Numbers

14.8 Forecasting: Final Thoughts

Chapter 14

So many topics, so little time …

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• A time-series variable (Y) consists of data observed over n periods of time.

• Businesses use time-series data - to monitor a process to determine if it is stable- to predict the future (forecasting)

• Time-series data can also be used to understand economic, population, health, crime, sports, and social problems.

Chapter 14

Time-Series Analysis

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• Time-series data are usually plotted as a line graph.

• Time is on the horizontal (X) axis.

• Trends and fluctuations are easier to see on a line graph.

• The following notation is used:

yt is the value of the time series in period t (t is an index denoting the time period t = 1, 2, …, n); n is the number of time periods; y1, y2, …, yn is the data set for analysis.

Time- Series Data

Chapter 14

Time-Series Analysis

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• To distinguish time-series data from cross-sectional data, use yt instead of xi for an individual observation.

Time-Series Data

Chapter 14

Measuring Time Series• Time-series data may be measured at a point in time.

• For example, prime rate of interest is measured at a particular point in time.

• Time-series data may also be measured over an interval of time.

• For example, gross domestic product (GDP) is a flow of goods and services measured over an interval of time.

Time-Series Components

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• The periodicity is the time interval over which data are collected.• Data can be collected once a year (e.g., 1 observation per year), quarter (e.g., 4

observations per year), month (e.g., 12 observations per year), etc.

Periodicity

Chapter 14

Additive versus Multiplicative Models• Time-series decomposition seeks to separate a time series Y into four

components:- Trend (T)- Cycle (C)- Seasonal (S)- Irregular (I)

• These components are assumed to follow either an additive or a multiplicative model.

Time-Series Components

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Additive versus Multiplicative Models

• The multiplicative model becomes additive if logarithms are taken (for nonnegative data):

Chapter 14

Time-Series Components

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• Trend (T) is the general movement over all years (t = 1, 2, ..., n).

• Trends may be steady and predictable, increasing, decreasing, or staying the same.

• A mathematical trend can be fitted to any data but may or may not be useful for predictions.

Trend

Chapter 14

Time-Series Components

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• Cycle (C) is a repetitive up-and-down movement about a trend that covers several years.

• Over a small number of time periods, cycles are undetectable or may resemble a trend.

Cycle

Chapter 14

Note: Forecasters generally ignore the cycle so the multiplicative model is just Y = T x S x I.

Time-Series Components

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• Seasonal (S) is a repetitive cyclical pattern within a year (or within a week, day, or other time period).

• Over a small number of time periods, cycles are undetectable or may resemble a trend.

• By definition, annual data have no seasonality.

Seasonal

Chapter 14

Time-Series Components

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• Irregular (I) is a random disturbance that follows no pattern.

• It is also called the error component or random noise reflecting all factors other than trend, cycle and seasonality.

Irregular

Chapter 14

Time-Series Components

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The main categories of forecasting models are:

Chapter 14

We will only look at this one category of models (more time to surf)

Trend Forecasting ML 3.2

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1. Make a nice Excel grapha. Highlight the data column (excluding heading)b. Insert > Line Chart (e.g., with markers)c. Add a descriptive chart title, etc.

2. Click on the line in the graph to select the variable.3. Right-click and choose Add Trendline.4. Select a trend (e.g., linear). Try several.5. Make forecasts (if desired).6. If quarterly or monthly data, calculate seasonal

factors (using MegaStat or Minitab).7. Multiply each numerical forecast by its seasonal

factor to get seasonally adjusted forecasts.

Steps in Forecasting:

Chapter 14

Trend Forecasting

Detailed examples follow…

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• Three trend models are especially useful in business applications:

Three Trend Models

• All three models can be fitted by Excel, MegaStat, or MINITAB.

Chapter 14

Linear Trend Model• The linear trend model has the form yt = a + bt

• It is the simplest and may suffice for short-run forecasting or as a baseline model.

Trend Forecasting

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Linear Trend Model

Chapter 14

Linear Trend Calculations• Linear trend is fitted by using ordinary least squares formulas.

• Note: Instead of using the actual time values (e.g., years), use an index xt = 1, 2, 3, …. as the independent variable.

Trend Forecasting

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Linear Trend Calculations

Chapter 14

Forecasting a Linear Trend• Once the slope and intercept have been calculated, a forecast can be

made for any future time period by inserting t = n+1, n+2, n+3, etc into the fitted trend equation.

these calculations are done by Excel (whew!)

Trend Forecasting

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• R2 can be calculated as

Linear Trend: Calculating R2

• An R2 close to 1.0 would indicate a good fit to the past data.

• However, a high R2 does not guarantee a good forecast. Projecting a trend assumes that nothing changes.

Chapter 14

Trend Forecasting

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• The exponential trend model has the form yt = aebt.

• Useful for a time series that grows or declines at the same rate (b) in each time period.

Exponential Trend Model

Chapter 14

Trend Forecasting

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• The exponential model (yt = aebt) is often used for data that may be assumed to grow at a steady percent growth rate (e.g., financial data).

• You can compare growth rates in two time-series variables with dissimilar data units by comparing their b estimates (i.e., the fitted growth rate b is unit-free)

• There may not be much difference between a linear and exponential model when the data set covers only a few time periods.

When to Use the Exponential Model

Chapter 14

• The linear model yt = a + bt and the exponential model yt = aebt are equally simple because they are two-parameter models, and a log-transformed exponential model is actually linear.

Trend Forecasting

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Calculations of the exponential trend are done by using a transformed variable zt = ln(yt) to produce a linear equation so that the least squares formulas can be used.

Exponential Trend Calculations

Chapter 14

Excel does all this. Once the least squares calculations are completed, Excel transforms the intercept back to the original units by exponentiation to get the correct intercept.

Caution: You can’t fit an exponential model if any data values are zero or negative.

Trend Forecasting

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• A quadratic trend model has the form yt = a + bt + ct2

• If c = 0, then the quadratic model becomes a linear model (i.e., the linear model is a special case of the quadratic model).

yt = a + bt + ct2

• Fitting a quadratic model is a way of checking for nonlinearity. If c does not differ significantly from zero, then the linear model would suffice.

Quadratic Trend

Chapter 14

Note: A quadratic equation is unfamiliar to many, and has no simple interpretation. Use it only when your data has a peak or trough and no other model suffices .

Trend Forecasting

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Depending on the values of b and c, the quadratic model can assume any of four shapes:

Quadratic Trend

Chapter 14

Note: Use the quadratic only for short term forecasts when no other model suffices .

Trend Forecasting

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Which Trend Model?

Chapter 14

Trend Forecasting

… or maybe none of the above will give reasonable forecasts

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“Fit” refers to how well the estimated trend model matches the observed historical past data. We usually look at R2 because it is familiar.

Five Measures of Fit

Chapter 14

Trend ForecastingWe usually refer to R2 because it is familiar.

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Example: Revenue of Amazon.com Inc (AMZN)• Eyeball the data – see anything unusual?

• Make a nice graph.

• Fit several trend models using Excel.

Chapter 14

Trend Forecasting

Quarter t y t Quarter t y t

03/31/2004 1 1530 03/31/2008 17 413506/30/2004 2 1387 06/30/2008 18 406309/30/2004 3 1462 09/30/2008 19 426412/31/2004 4 2541 12/31/2008 20 670403/31/2005 5 1902 03/31/2009 21 488906/30/2005 6 1753 06/30/2009 22 465109/30/2005 7 1858 09/30/2009 23 544912/31/2005 8 2977 12/31/2009 24 952003/31/2006 9 2279 03/31/2010 25 713106/30/2006 10 2139 06/30/2010 26 656609/30/2006 11 2307 09/30/2010 27 756012/31/2006 12 3986 12/31/2010 28 1294703/31/2007 13 3015 03/31/2011 2906/30/2007 14 2886 06/30/2011 3009/30/2007 15 3262 09/30/2011 3112/31/2007 16 5672 12/31/2011 32

Objective: Fill in these 4 boxes

Revenue is in $millions (e.g., first data value is 1.53 billion)

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Example: Revenue of Amazon.com Inc (AMZN)

Chapter 14

Trend Forecasting

Make nice graph, then click on the data series Be sure to

click these 2 boxes

Excel’s “Polynomial Order 2” is a “Quadratic” trend

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Chapter 14

Example: Amazon Revenue

y = 284.58x + 117.77R² = 0.7396

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Note: Excel will show forecasts on the graph but no numbers are given.

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Fitted Trends: Amazon Revenue

Chapter 14

If necessary, format the fitted trend label (right-click it) to show more decimals.

Example: Amazon Revenue

Moving average (not really a trend model)

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Chapter 14

Example: Amazon Revenuey = 284.58x + 117.77

R² = 0.7396

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y = 12.752x2 - 85.227x + 1966.8R² = 0.8168

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y = 1316.5e0.0686x

R² = 0.8861

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Interpretation: growing at $284.58 million per quarter, 74% of variation explained by linear trend model, forecasts seem low?

Interpretation: complex nonlinear equation, 82% of variation explained by quadratic trend

Interpretation: growing 6.86% per quarter, 89% of variation explained by exponential trend, believable forecasts

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Chapter 14

Example: Amazon Revenuey = 284.58x + 117.77

R² = 0.7396

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y = 12.752x2 - 85.227x + 1966.8R² = 0.8168

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R² = 0.8861

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Excel formula for t = 32 forecast: =284.58*32 + 117.77

Excel formula for t = 32 forecast: =1316.5*EXP(.0686*32)

Excel formula for t = 32 forecast: =12.75*32^2 - 85.227*32 + 1966.8

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Chapter 14t Linear Quad Expon

29 8370 10220 962530 8655 10887 1030931 8939 11579 1104132 9224 12298 11825

Unadjusted Forecasts

t Linear Quad Expon29 =117.11+284.58*J11 =1966.8-85.227*J11+12.752*J11 2̂ =1316.5*EXP(0.0686*J11)30 =117.11+284.58*J12 =1966.8-85.227*J12+12.752*J12 2̂ =1316.5*EXP(0.0686*J12)31 =117.11+284.58*J13 =1966.8-85.227*J13+12.752*J13 2̂ =1316.5*EXP(0.0686*J13)32 =117.11+284.58*J14 =1966.8-85.227*J14+12.752*J14 2̂ =1316.5*EXP(0.0686*J14)

Unadjusted Forecasts

Note: In this example, the time index t = 29, 30, 31, 32 is in cells J11, J12, J13, J14

Note: To make these forecasts, the formulas from fitted trends were entered into cells beside the time index t = 29, 30, 31, 32 (as shown below).

Example: Amazon Revenue

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Chapter 14

Example: Amazon Revenue

to make future forecasts, insert formula in each cell for all three fitted models using time index in column A (or wherever it is)

Note: The year and quarter are just labels – they are not used in any of the calculations.

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Chapter 14t Linear Quad Expon

29 8370 10220 962530 8655 10887 1030931 8939 11579 1104132 9224 12298 11825

Unadjusted Forecasts

Comment: The linear forecasts are much more conservative than the other two trend models. Quadratic forecasts are the most aggressive, though only slightly more than the exponential forecasts.

Comment: These are quarterly data, so now we should adjust the forecasts for seasonality.

Example: Amazon Revenue

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Four Trend-Fitting Criteria

Criteria for selecting a trend forecasting model:

Criterion Ask Yourself• Occam’s Razor Would a simpler model

suffice?• Overall fit How does the trend fit the

past data?• Believability Does the extrapolated trend

“look right”?• Fit to recent data. Does the fitted trend match the

last few data points?

Chapter 14

Trend Forecasting

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• When the data periodicity is monthly or quarterly, calculate a seasonal index and use it to deseasonalize the data.

• For the multiplicative model, a seasonal index is a ratio.

• The seasonal indexes must sum to 12 for monthly data or to 4 for quarterly data.

• In a multiplicative model, seasonal indexes near 1.00 suggest a lack of seasonality:

Y = T x S x I if S = 1.00 then S disappears

When and How to Deseasonalize

Chapter 14

Forecasting with Seasonality ML 3.3

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Step 1: Calculate a centered moving average (CMA) for each month (quarter).

Step 2: Divide each observed yt value by the MA to obtain seasonal ratios.

Step 3: Average the seasonal ratios by the month (quarter) to get raw seasonal indexes.Step 4: Adjust the raw seasonal indexes so they sum

to 12 (monthly) or 4 (quarterly).Step 5: Divide each yt by its seasonal index to get

deseasonalized data.

When and How to Deseasonalize

Chapter 14

Seasonality

Note: We rely on MegaStat or another computer package for these complex calculations.

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MegaStat Menus

Chapter 14

Seasonality

to label the data by year and quarter

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MegaStat’s Seasonal Analysis

Chapter 14

Seasonality

CenteredMoving Ratio to Seasonal Net sales (millions)

t Year QuarterNet sales (millions) Average CMA Indexes Deseasonalized1 2004 1 1,530 0.956 1,601.1 2 2004 2 1,387 0.837 1,657.8 3 2004 3 1,462 1776.737 0.823 0.851 1,717.7 4 2004 4 2,541 1868.901 1.360 1.356 1,874.0 5 2005 1 1,902 1964.049 0.968 0.956 1,989.9 6 2005 2 1,753 2067.995 0.848 0.837 2,094.8 7 2005 3 1,858 2169.625 0.856 0.851 2,182.3 8 2005 4 2,977 2265.000 1.314 1.356 2,195.6 9 2006 1 2,279 2369.375 0.962 0.956 2,384.3

10 2006 2 2,139 2551.625 0.838 0.837 2,556.0 11 2006 3 2,307 2769.750 0.833 0.851 2,709.6 12 2006 4 3,986 2955.125 1.349 1.356 2,939.7 13 2007 1 3,015 3167.875 0.952 0.956 3,154.3 14 2007 2 2,886 3498.000 0.825 0.837 3,448.7 15 2007 3 3,262 3848.750 0.848 0.851 3,831.3 16 2007 4 5,672 4135.875 1.371 1.356 4,183.2 17 2008 1 4,135 4408.250 0.938 0.956 4,326.1 18 2008 2 4,063 4662.500 0.871 0.837 4,855.2 19 2008 3 4,264 4885.750 0.873 0.851 5,008.2 20 2008 4 6,704 5053.500 1.327 1.356 4,944.3 21 2009 1 4,889 5275.125 0.927 0.956 5,114.9 22 2009 2 4,651 5775.250 0.805 0.837 5,557.8 23 2009 3 5,449 6407.500 0.850 0.851 6,400.0 24 2009 4 9,520 6927.125 1.374 1.356 7,021.1 25 2010 1 7,131 7430.375 0.960 0.956 7,460.5 26 2010 2 6,566 8122.625 0.808 0.837 7,846.2 27 2010 3 7,560 0.851 8,879.4 28 2010 4 12,947 1.356 9,548.5

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MegaStat’s Seasonal Indexes

Chapter 14

Seasonality

Calculation of Seasonal Indexes1 2 3 4

2004 0.823 1.3602005 0.968 0.848 0.856 1.3142006 0.962 0.838 0.833 1.3492007 0.952 0.825 0.848 1.3712008 0.938 0.871 0.873 1.3272009 0.927 0.805 0.850 1.3742010 0.960 0.808

mean: 0.951 0.833 0.847 1.349 3.980adjusted: 0.956 0.837 0.851 1.356 4.000 <--- adjusted so they add to 4.000

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MegaStat’s Graph

Chapter 14

Seasonality

Note: MegaStat’s graph does not show any forecasts (only the deseasonalized time series

So … we have to make our own numerical forecasts

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Seasonal Adjustment

Chapter 14

Seasonality

0.956 Qtr 10.837 Qtr 20.851 Qtr 31.356 Qtr 4

MegaStat Seasonals

Now, multiply each trend forecast by its quarterly seasonal factor

t Linear Quad Expon29 8370 10220 962530 8655 10887 1030931 8939 11579 1104132 9224 12298 11825

t Linear Quad Expon29 8018 9790 922130 7215 9076 859431 7607 9854 939532 12520 16693 16051

Unadjusted Forecasts

Seasonally Adjusted Forecasts

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Compare Forecasts and Choose One

Chapter 14

Seasonality

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More Time Series Methods ML 3.4

Chapter Contents14.1 Time-Series Components14.2 Trend Forecasting14.3 Assessing Fit14.4 Moving Averages14.5 Exponential Smoothing14.6 Seasonality14.7 Index Numbers14.8 Forecasting: Final Thoughts

Chapter 14

… when no trend model works

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Moving Averages

• In cases where the time series y1, y2, …, yn is erratic or has no consistent trend, there may be little point in fitting a trend line.

• A simple approach is to calculate either a trailing or centered moving average.

Trendless or Erratic Data

Chapter 14

Trailing Moving Average (TMA)• The TMA simply averages the data over the last m periods.

• The TMA smoothes the past fluctuations in the time series in order to see the pattern more clearly.

• Choosing a larger m yields a “smoother” TMA but requires more data.

Note: Excel uses the TMA method.

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• The CMA smoothing method calculates the mean of the current observation and observations on either side of the current data. For example, for m = 3:

Centered Moving Average (CMA)

Chapter 14

• When m is odd (m = 3, 5, etc.), the CMA is easy to calculate.• When m is even, the mean would lie between two data points and

would not be correctly centered, so we would take a double moving average.

Moving Averages

Caution: Excel does not offer the CMA method (only TMA).

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Exponential Smoothing

• The exponential smoothing model is a special kind of moving average.

• This one-period-ahead forecasting technique is utilized for data that have up-and-down movements but no consistent trend.

• The updating formula iswhere

Forecast Updating

Chapter 14

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• The forecast Ft+1 is a weighted average of yt (the current data) and Ft (the previous forecast).

• The value of (the smoothing constant) is the weight given to the latest data.

• A small value of would give low weight to the most recent observation.• A large value of would give heavy weight to the previous forecast.• The larger the value of , the more quickly the forecasts adapt to recent

data.

Smoothing Constant ()

Chapter 14

Choosing the Value of • If = 1, there is no smoothing at all and the forecast for the next

period is the same as the latest data point.

Exponential Smoothing

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• Where do we get the initial forecast F1 (i.e., how do we initialize the process)?

• Method AUse the first data value. Set F1 = y1

• Although simple, if y1 is unusual, it could take a few iterations for the forecasts to stabilize.

Initializing the Process

Chapter 14

• Method BAverage the first six data values. Set

F1 = (y1 + y2 + y3 + y4 + y5 + y6)/6• This method consumes more data and is still somewhat vulnerable to

unusual y-values.

Exponential Smoothing

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• The effect of past data diminishes as time increases.

• To see this, replace Ft with Ft 1 and repeat this type of substitution indefinitely to obtain

• The coefficients diminish so older yt values have less effect on the current forecast.

Effect of Past Data

Chapter 14

• Note that Ft 1 depends on Ft, which in turn depends on Ft 1, and so on all the way back to F1.

Exponential Smoothing

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Example from LearningStats:

Chapter 14

Exponential Smoothing

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• The Box-Jenkins method uses several different types of time- series modeling techniques that fall into a class called ARIMA (Autoregressive Integrated Moving Average) models

• AR (autoregressive) models take advantage of the dependency that may exist between values in the time series.

• MA (moving average) models take advantage of the dependency that may exist between errors in the forecasts.

• Although they are more powerful and general than trend models, these methods require sophisticated software and additional training. Excel and MegaStat do not offer these methods.

Chapter 14

Advanced Methods

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Assignments ML 3.5

• Project P-1 (data, tasks, questions)• Review instructions• Look at the data• Your task is to write a nice, readable report (not a spreadsheet)• Length is up to you

• Project P-2 (data, tasks, questions)• Review instructions• Look at the data• Your task is to write a nice, readable report (not a spreadsheet)• Length is up to you

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Projects: General InstructionsC

hapter 0

General Instructions

For each team project, submit a short (5-10 page) report (using Microsoft Word or equivalent) that answers the questions posed. Strive for effective writing (see textbook Appendix I). Creativity and initiative will be rewarded. Avoid careless spelling and grammar. Paste graphs and computer tables or output into your written report (it may be easier to format tables in Excel and then use Paste Special > Picture to avoid weird formatting and permit sizing within Word). Allocate tasks among team members as you see fit, but all should review and proofread the report (submit only one report).

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You will be assigned a company and team members (see Moodle). Delegate tasks and collaborate as seems appropriate, based on your various skills. Submit one report. Data: Download your company’s quarterly revenue data 2006-2012 from Moodle or from Doane’s teaching web page. Analysis: (a) Briefly describe the company’s history, products, services, competition, and market conditions (e.g., Yahoo or Mergent or Google). (b) Fit several trends (e.g., linear, quadratic, exponential) using Excel. (c) Interpret each fitted trend equation. Discuss and compare the R2 statistics. (d) Forecast the next 4 quarters (t = 29, 30, 31, 32) based on trend alone using each fitted trend model (i.e., plug in the time index for periods n+1, n+2, n+3, n+4). (e) Use MegaStat (or Minitab) to calculate quarterly seasonal factors. Is there noticeable seasonality? (f) Ambitious students: Multiply each quarterly trend forecast by its seasonal factor. Discuss the effect of the seasonal adjustment. (g) Using the four criteria for assessing forecasts (see p. 614), which trend model (if any) would yield credible forecasts? If none, then what?

Project P-2

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Project P-2

Note: Use the “cleaned”: spreadsheets for your forecasting project. The others are the “raw data” from Mergent Online.

A Few ExamplesThese are from Doane’s teaching web page (only the “cleaned” files were posted to Moodle). You can look up your company’s stock ticker symbol on the internet (e.g., EMC)

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Note: Use only the “cleaned”: spreadsheets for your P-2 project. The others are “raw data” from Mergent Online.

Project P-2

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Project P-2

Hint: Watch the instructor’s video walkthrough using Amazon’s revenue as an example (posted on Moodle)