Module 4: Time Series Decompositon Flashcards
A time series can be decomposed into four components what are the four components?
- Trend component (Tt)
- Cyclical Component (ct)
- Seasonal Component (st)
- Irregular components (It)
What is the irregualr component also known as?
Residual
How do we link these componets togethre?
Through two time series model:
1. Additive model
2. Mutiplivative model
How do the models differ?
- The models differ from each other in the way each component affects xt
What is the addictive models equation?
The equation was
Xt = Tt + Ct + St + It
What is the multiplicative model?
Xt = Tt * Ct * St * It
How does the addictive model affect Xt?
In the additive model, the effect of any component on Xt is not related to the value of the other components
Example: If Ct incerases by 1 what happens to Xt and the other components?
If Ct increases by 1, then Xt also increases by 1 no matter what the values of TT, ST, It
How does the multiplicative model affect Xt
In the multiplicative model, the effect of any component on Xt also depends on the value of others
In this course, we focus mostly on the addictive model however when shall we look at the multiplicative model?
The only time the multiplicative model will be preferred is when the series is expressed in logs because it implies the following addictive model:
Log(xt) = log(tt* CTStIt)
= log(tt) + log(St) + log(Ct) + log(It)
How are the four components ordered in?
Their are in order with respect to their frequency going from the very low frequency (TT) to very high frequency (IT)
What is frequency?
– To understand frequency lets consider regular waves
- Frewuency of a wave allows us to see how many times it takes for a wave to repeat itself in a certain amount of time
What is low frequency?
- The frequency of a wave is low when it takes a long time to complete one cycle (so we won’t see many waves)
What is a high frequency?
- if a wave has a high frequency it takes less time for a wave to repeat itself.
- Therefore we see many high frequency waves in a given time period
List the components in order from low frequency to high
- Trend –> very low frequency
- Cyclical–> medium low
- Seasonal –> medium high
- Irregualr –> very high
How does the trend component look like in a wave?
- We see usually a fraquation of way a the frequency is so low so it looks more like a straight line than a wave
how does the cyclical wave look like?
it look like a wave but its frequency is lower than the two other’s above and the other two are higher in order of repeats in wave
What is the low frequency component?
(Tt+Ct) –> for the addicitive model
(Tt*Ct) –> for the multiplciative model
What is the high freqnecy component
–> (st+ It) –> addictive model
–> (st*it) –> multiplicative model
How to make a component more visible?
- they become more visible when the lower frequency components have been removed
- This suggests that the components should be removed one by one starting with low (tt) and ending with high (It)
If I remove ‘tt’ from ‘xt’ what happens?
The next component becomes visible:
Ct
if I removed tt from xt what is it called?
The detrended series
When we remove components from Xt what is this called?
DECOMPOSITION
How to look at a trend series? [when in its graph form]
If its increasing over the period as it looks most like a straight line tis positive
How to look at a cycle series?
- A positive value from C indicates that the series is above the trend , negative when below and it is on the trend when Ct is equal to zero
How to look at a Seasonal series?
- A positive St means that that the series are higher than the low frequency component on average and negative inversely
Why is it called seaonal
- It is the part of Xt that is affected by the seasonal however it does not just mean to just seasons.
- This is effect that happens in each month, quarter (whatever happens)
WHat should we remember about trend + cycle
It’s based on what we see as theoretical meanings might vary to what were saying but its based o the visualization
What can we consider the defintion of a trend to be?
In this course we will assume that the trend is a linear or a quadratic function of time
How can a linear trend be written as?
Tt = a+ Bt
- We assume that the time unit t is one year
- we also assume b is the annual change
For the time we can either use:
1. The regular date (1990) or the numerical value of the date
What’s something to note about figuring out which series best fits the data
- We have to choose between if our graph best fits the linear or quadratic component.
- If in the data it does not seem a linear best fits choose quadratic trend
What is a quadratic trend?
Its the alterantive to a straight line:
Tt = a+bt + ct^2
What does the T^2 add to the trend
It adds curavtire to the trend and the type of curvature is dependant on the value of C.
How to figure out what the trend increases by?
- Find the TT for both
- Subtract both from each other
If there’s a positive co-efficent of T^2 what does that imply?
It means that the trend is increasing.
How to know if a series with an icerasing trend may be linear in the log-scale?
If the growth rate is constatnt from both years.
How to see if the growth rate is constant
- You have the trend equation (for a linear trend) for two years
- Calculate the amounts and subtract t - t^1
What if we can see the growth rate is not constant but increasing?
- We can tell its not increasing when we failed to properly remove the trend as its still ushaped
- If we want to keep the logscale we have to fit a quadratic trend to the log of co2
What is the ‘cycle’ (ct) component?
It has a higher frequency than the trend but lower frequency than seasonal and irregular
How can we extract Ct?
We can apply a trasnformation that elimnates the components with higher frequency.
When we apply a trasnfromation that eliminates the components with a higher frequency what is that called?
Smoothing
Why is smoothing important?
It removes high flucutuations.
In this course what is the ‘simple smoothing’ technique we will be using?
This is called moving average (MA)
What is the moving average?
The process is every observations is replaced by the average points around it
Explain how we do the centered moving average of 5?
Xt-2 + xt -1 + xt + xt+1 + xt+2 / 5
This would replace the 3rd observation
Explain the moving average of 13
Its 6 points bfr Xt and 6 after
This would replace 7th observation
What’s something to note we do with MA?
- We only consider ‘odd orders’
- Loose observations at the beginning and end
For monthly or quarterly series what should we put the MA?
Monthly = 13
Quarterly = 5
How to interpret the ‘cycle’ component?
- This means on average Ct = 0 the Xt is equal to the trend
- CT > 0 it means that it is above the trend
- CT <0 it means its below
What is a seasonality component?
This is a flcutuation that is influenced by either month, or quarter depending on the frequency of the data
How to obtain seasonal component?
- Have the monthly average (whether it be calculated or not)
- Take the monthly average of the data and calculate the monthly average of the entire year
- Subtract the At with average(year)
How to understand the seaosnal compoent (what the trend is)
- When St = 0 which is the average of all st there is no seasonal effect
- if its positive the series in above (CT+tt) on average and below when St is negativee
The addictive model each component has teh same unit as xt how do we interpret it? –> summarization
(𝑋𝑡−𝑇𝑡)
(X t−T t): A zero value means that the observation is equal to the trend, a positive value means that the observation is higher than the trend and inversely for negative values.
𝐶t : A zero value means that the observation is equal to the trend on average, a positive value means that the observation is higher than the trend on average and inversely for negative values. In this context, on average means when the high frequency component
𝑆𝑡+𝐼t is equal to 0.
𝑆𝑡: A zero value means that the observation is equal to the low frequency component
t𝑡+𝐶𝑡 on average, a positive value means that the observation is higher than 𝑇𝑡+𝐶𝑡 on average and inversely for negative values. Here, on average means when the very high frequency component It is equal to zero
𝐼t :Interpreting this component is of no interest in this course.
The multiplicative model interpretations
[Log(Xt) - Log(tt)] this is difference in logs which implies that it measures the ‘differnce’ between Xt and Tt in percentage
Ex: if its 3% it means that Xt is 3% higher than TT and negative is lower
Log(Ct) : It has the same interpretation as Log(xt)- log(tt) but we need to ‘add’ on average
: Example that 3% is on average
–> here on average means when the log of the high frequency component (ST*It) is equal to zero
Log(st) –> this is equal to [log(xt) - log(tt*Ct)
Therefore the percentage difference between xt and the low frequency component on average
- On average means when the log of the very high frequency component It is equal to zero
Log(it) –>no interpretations
How to fit a linear trend to a monthly series in excel
- We are basically trying to find the coefficants of and b that fit the long term behavior of our series
2.The process of finding the equation that best fit data points is called regression
- Go to excel
- Select data analysis
- Select data menu
- Select regression hit play
- In the y range bar we put data points
- In the x input range bar we put the ‘dates in decimal format’
- Hit okay and here you will see a table full of numbers you only need to look where coefficants are and hit the x variable 1
hwo to create the trend component
- Craete a new column called ‘linear trend’
- To generate the trend we need to perform the operation A+b*t
- Remember to freeze a and B –> F4 shortcut
How to fit a qudaryic trend to series?
- We need to cmpute the three coefficants in the following equation –> A+bt+ct^2
- T^2 is the square value of decimal data
- To obtain the three coefficants
- Select data analysis
- Hit regression
- Do not chang y values but change x and make sure it includes both decdate and decdate^2 (make sure you bput both columns next to each other)
- Make a column for quadratic trend
- Do A+Bt+CT^2 (freeze a, b ,c)
How to express our series in logs?
- We sumply apply the same trasnformed series but change the data and convert to logs
- Use LN
How to extract the cycle component?
- It can be extracted from the dterended series (xt -tt)
- To extract the cycle component we have to use the MA(13) if monthly to the detrended series
- For a monthly time series we cannot compute the first and last 6 observations
How to compute St (Seasonality)
- To compute ‘St’ we first compute the average of (St+It) = High frequency month by month after subtarct the annual average
What is the process of removing a seasonal component called?
Seasonal adjustment or deconlization
[Co2 - seasonal]
How to extract the seasonal component of a series?
- Insert a column and name it [High frequency] = (St + It) = (Xt - Tt- Ct) or (detrended - cycle]
- Delete the content of the first and last 6 cells of theHigh frequency column
- Insert two columns and name it S0 and Seasonal
- For S0 we use average if
{Average if has three components] - Put the first variable of the period
- Highlight the entire period column [month, quarter]
- Select and highlight all of [high frequency]
- Then subtact S0 - average[entire period of S0 for 1] so if its a 12 months take all 12 months and if its for 1 year take the 4 quarters and freeze it for each one
What is a purpose for deconalizing?
Its to illusrate the comovement between to series at different frequencies
What can we consider the most macroeconomic time series?
Estimates which means they are not ‘exact measures’
What is an unbiased error
A measurement error is unbiased if it is equal to zero on average
What can unbiased measurement error likely effect
it likely only affects the high-frequency components of a time series
What is a biased error?
A measurement is biased if it is not zero on average
