time series models

Question 1

exponential smoothing equation

Answer 1

st - the baseline at time period t
xt- the observed vlaue (response)

st = axt + (1-a)st-1

0<a<1

Question 2

exponential smoothing tradeoff

Answer 2

a -> 0 there is a lot of randomness in the system, fluctuations are due to randomness, therefore yesterdays baseline is probably a good indicator of todays baseline
a-> 1 not much randomness in the system, fluctuations are probably due to changes in baseline

Question 3

How to start exponential smoothing?

Answer 3

S1 or baseline = x1 or first value recorded

Question 4

What does exponential smoothing not deal with?

Answer 4

-doesn’t deal with trends or cyclical variations

Question 5

trends

Answer 5

the value is increasing or decreasing over time

Question 6

What is time series data?

Answer 6

that in which the same response is known for many time periods

Question 7

cyclical patterns

Answer 7

ex - annual temp cycles
-weekly sales cycles
-daily blood pressure cycles

Question 8

add trend to exp smoothing calculation

Answer 8

Tt: trend at time period t

st = alphaxt + (1-alpha)(st-1+Tt-1)

Question 9

How do you calculate trend?

Answer 9

just like you do for the baseline

Tt = Beta(St- St-1) + (1-Beta)Tt-1

Question 10

Trend initial condition

Answer 10

T1 (trend) = 0

Question 11

What two methods can you use to deal with cyclical patterns?

Answer 11

additive and multiplicative

Question 12

Seasonalities additional variables

Answer 12

L: length of cycle
Ct: the mutiplicative seasonality for time t
(inflate or deflate the observed observation)

Question 13

Baseline formula w/ trend and seasonality (multiplicative)

Answer 13

st = alphaxt/Ct-L + (1-alpha)(st-1+Tt-1)

Question 14

How do we update seasonality? What is the initial value?

Answer 14

Ct = gamma(xt/St)+ (1-gamma)Ct-L
-no initial cyclic effect b/c it can’t be measured until the end of the first season

Question 15

How would you interpret a seasonality value of 1.1 for weekly cyclical data? How would this change your results?

Answer 15

on that day, the value is 10% higher just because it is that day.

if you sold 550 items, 500 was your baseline and 50 was because it was sunday

Question 16

Multiplicative seasonality starting condition

Answer 16

-start by multiplying by 1(no seasonality/cyclic effect) for the first L values of C

Question 17

exponential smoothing also sometimes called

Answer 17

single, double or triple exponential smoothing depending on how many aspects like trend and seasonality you include

Question 18

triple exponential smoothing with base equation plus trend and seasonality is also called?

Answer 18

winters method or holt-winters

Question 19

What does exponential smoothing do?

Answer 19

peaks and valleys are smoothed out

st = alphaxt + (1-alpha)st-1

ex- alph = 1/2

when xt is high, St is not as high, its pulled down by (1-alpha)st-1
when xt is low, St is not as low, it’s pulled up by (1-alpha)st-1

Question 20

Why is exponential smoothing exponential?

Answer 20

St-1 can be rewritten as

st-1 = alphaxt-1 + (1-alpha)st-2

and when you substitute the value, (1-alpha) multiplies the St-1 substitution so that you get (1-alpha )alpha xt-1 + (1-alpha )^2 St-2
and so on, all the way back to the first value in the time series

Question 21

Which time periods contribute to current baseline estimate? Which time periods contribute the most?

Answer 21

Every past observation, all the previous data is baked in to St-1

The more recent time periods contribute the most, because we take (1-alpha)^exponent that increases by 1 with each time period

Question 22

exponential smoothing forecasting baseline

Answer 22

original equation - st = alphaxt + (1-alpha)st-1

prediction

st+1 = alphaxt+1 + (1-alpha)st

xt+1 is unknown, so our best guess is St or the previous periods baseline

doing that calculation we get our forecast…

Ft+1 = alphaSt + (1-alpha)st, so Ft+1 = St

Ft+k = St, k = 1,2,…. the estimate remains the same for all future time periods

forecast error gets higher farther into the future

Question 23

exponential smoothing forecasting including trend

Answer 23

we just include the trend

the best estimate of the next baseline is our current baseline estimate
best trend estimate is our most current trend estimate

therefore
our forecast for time t+1 is
Ft+1 = St+Tt

the trend is the same going forward

Question 24

exponential smoothing forecasting including trend and multiplicative seasonality

Answer 24

the best estimate of the next time periods seasonal factor is

Ct+1 = CT(t+1) - L or the multiplicative seasonality the last cycle at this time

Forecast
Ft+1 = (St+Tt) C(T+1) - L and remains the same going forward

Question 25

3 key parts of ARIMA

Answer 25

1-Differences
if the data is not stationary (trend or seasonality), differences in the data might be stationary

2-Autoregression - predicting current value based on previous time periods values

3-Moving Average - previous errors as predictors

Question 26

What does ARIMA stand for?

Answer 26

autoregressive integrated moving average

Question 27

What does it mean for data to be stationary?

Answer 27

mean variance, and other measures are all expected to be constant over time

Question 28

Types of differences

Answer 28

First order differences - differences of consequtive observations
2nd order - difference of differences
3rd order - differences of differences of differences

or the dth order difference (infinite differecnes of differences of differences…)

Question 29

types of auto regression

Answer 29

order infitinity regressive model- exp smoothing uses autoregression because we’re making predictions based on the same value
-uses data as far back as we have

order p autoregressive - go back p timeperiods

Question 30

autoregression meaning breakdown

Answer 30

regression - predicting the value based on other factors

auto
-instead of using other factors to predict we use earlier values of what we’re measuring to predict
-only works with time series data

Question 31

What does arima do?

Answer 31

it combines autoregression and differencing

-autoregression on the differences
-use p time periods of pervious observations to predict the dth order differences

Question 32

Moving avg part of arima components

Answer 32

previous errors as predictors
Et =(xhatt - xt)

order -q moving avg
go back q time periods

Question 33

Arima model

Answer 33

arima pdq model

D(d)t = avg + sum pth order autoregression on dth order differences - sum of qth order moving avg
-dth order differences
-pth order autoregression
-qth order moving avg

statistical software can find pdq

Question 34

Arima model equivalence

Answer 34

specific values of PDQ give other more basic models

arima(0,0,0)
-white noise ( no patterns)
arima(0,1,0)
-random walk
arima(p,0,0)
-autoregressive
arima(0,0,q)
-moving avg
arima(0,1,1)
-basic exponential smothing

Question 35

Can arima be used for short term forecasting like exponential smoothing?

Answer 35

short term forecasting
-better than exp. smoothing
-when the data is more stable with fewer peaks valleys and outliers
usually need

need about 40 past data points for data to work well

Question 36

What does GARCH stand for?

Answer 36

generalized autoregressive conditional heteroscedasticity

Question 37

What does garch do?

Answer 37

estimate or forecast the variance of something

Question 38

what does variance do?

Answer 38

estimate the amount of error in our estimate

Question 39

Why is variance estimation important in finance?

Answer 39

investment
-traditional portfolio optimization model balances expected return of a set of investments with the amount of volatility

tradeoff between return and risk

variance - proxy for volatility or risk

Question 40

Differences between Garch and Arima

Answer 40

garch uses

-varinaces/squared errors, not observations/ linear errors
-raw variances - not differences of variances like we use differences in arima

-otherwise they’re very similar

Question 41

3 methods for analyzing time series data

Answer 41

-exp smoothing
-arima
-garch