Fundamentals of Time Series Econometrics
E_MFAE_FTSE
,Table of contents
Week 1 ................................................................................................................................................................................................ 3
Week 2 ................................................................................................................................................................................................ 6
Week 3 .............................................................................................................................................................................................. 10
Week 4 .............................................................................................................................................................................................. 13
Week 5 .............................................................................................................................................................................................. 17
Week 6 .............................................................................................................................................................................................. 20
, Week 1
Deterministic – can be determines by past.
However, no phenomena is deterministic, because unknown factors.
Stochastic Process – it’s a mathematical model which explains, not one random event, but a random event, but rather a whole
sequence of random variables {𝑋𝑡 } ordered in time.
{𝑋𝑡 } – This is a mathematical notation for a sequence of random variables indexed in time.
• Realization
o Single Realization – one run of process o Multiple realizations – many runs of the
one path same process, different paths due to
randomness.
o
o
Time Series components:
• Trend
o Long-term/short-term growth.
• Seasonality
o Always fixed and known period.(time of the year, day of the weak)
• Cyclical
o Always due to economic factors, not fixed periods
Time series can be decomposed into its componenents/
• Additive: 𝑋𝑡 = 𝑇𝑡 + 𝑆𝑡 + 𝑅𝑡 – in items.
• Multiplicative: 𝑋𝑡 = 𝑇𝑡 × 𝑆𝑡 × 𝑅𝑡 – percentages.
o 𝑋𝑡 – in time T.
o T – trend cycle component
o 𝑅𝑡 - unpredictable
o S – seasonal
o log 𝑋 = log 𝑇 × 𝑆 × 𝑅 → log 𝑋 = log 𝑇 + log 𝑆 + log 𝑅 – Logarithms are often used to transform multiplicative
models into additive models
Methods of decomposition
• Classical Decomposition
• X-11 Decomposition
• STL Decomposition
Classical Decomposition
• Trend Cycle computed my moving average.
1
o m-MA: 𝑇̂𝑖 = ∙ ∑𝑘𝑗=−𝑘 𝑥𝑡+𝑗 , where 𝑚 = 2𝑘 + 1
𝑚
▪
• Seasonal component: computed by averaging the deviations from the trend-cycle each season
• Remainder Component: the residuals after removing the trend-cycle and seasonal component.
E_MFAE_FTSE
,Table of contents
Week 1 ................................................................................................................................................................................................ 3
Week 2 ................................................................................................................................................................................................ 6
Week 3 .............................................................................................................................................................................................. 10
Week 4 .............................................................................................................................................................................................. 13
Week 5 .............................................................................................................................................................................................. 17
Week 6 .............................................................................................................................................................................................. 20
, Week 1
Deterministic – can be determines by past.
However, no phenomena is deterministic, because unknown factors.
Stochastic Process – it’s a mathematical model which explains, not one random event, but a random event, but rather a whole
sequence of random variables {𝑋𝑡 } ordered in time.
{𝑋𝑡 } – This is a mathematical notation for a sequence of random variables indexed in time.
• Realization
o Single Realization – one run of process o Multiple realizations – many runs of the
one path same process, different paths due to
randomness.
o
o
Time Series components:
• Trend
o Long-term/short-term growth.
• Seasonality
o Always fixed and known period.(time of the year, day of the weak)
• Cyclical
o Always due to economic factors, not fixed periods
Time series can be decomposed into its componenents/
• Additive: 𝑋𝑡 = 𝑇𝑡 + 𝑆𝑡 + 𝑅𝑡 – in items.
• Multiplicative: 𝑋𝑡 = 𝑇𝑡 × 𝑆𝑡 × 𝑅𝑡 – percentages.
o 𝑋𝑡 – in time T.
o T – trend cycle component
o 𝑅𝑡 - unpredictable
o S – seasonal
o log 𝑋 = log 𝑇 × 𝑆 × 𝑅 → log 𝑋 = log 𝑇 + log 𝑆 + log 𝑅 – Logarithms are often used to transform multiplicative
models into additive models
Methods of decomposition
• Classical Decomposition
• X-11 Decomposition
• STL Decomposition
Classical Decomposition
• Trend Cycle computed my moving average.
1
o m-MA: 𝑇̂𝑖 = ∙ ∑𝑘𝑗=−𝑘 𝑥𝑡+𝑗 , where 𝑚 = 2𝑘 + 1
𝑚
▪
• Seasonal component: computed by averaging the deviations from the trend-cycle each season
• Remainder Component: the residuals after removing the trend-cycle and seasonal component.
