egzamin Flashcards
1
Q
Czym jest szereg czasowy?
A
Zbiór obserwacji statystycznych uporządkowany według czasu.
2
Q
Jakie są dwa typy szeregów czasowych?
A
- Szeregi momentów
- Szeregi okresów
3
Q
Czym charakteryzują się szeregi czasowe momentów?
A
Zawierają informacje o poziomie zjawiska w wyróżnionych momentach czasu.
4
Q
Czym charakteryzują się szeregi czasowe okresów?
A
Zawierają dane dotyczące kształtowania się poziomu danego zjawiska w całym okresie przyjętym za jednostkę czasu.
5
Q
Jakie elementy finansowe najczęściej bada się w szeregach czasowych?
A
- Ceny akcji
- Wartości indeksów giełdowych
- Stopy procentowe
- Kursy walutowe
- Ceny towarów z giełd towarowych
6
Q
Czym jest przyrost absolutny?
A
Różnica w poziomie zjawiska mierzonego w dwóch różnych momentach lub okresach.
7
Q
Jak definiuje się stopę zwrotu?
A
Jako tempo zmian, które definiuje różnicę poziomu zjawiska w badanym okresie w porównaniu do okresu podstawowego.
8
Q
Jakie są podstawowe miary opisowe w analizie finansowych szeregów czasowych?
A
- Przeciętny poziom zjawiska
- Zmienność
- Miary symetrii
- Koncentracja
- Spłaszczenia
9
Q
Czym jest proces stochastyczny?
A
Rodziną zmiennych losowych określonych w przestrzeni probabilistycznej.
10
Q
Co oznacza, że proces stochastyczny jest stacjonarny?
A
Średnia i wariancja nie zmieniają się w czasie, a kowariancje zależą tylko od odstępu między momentami obserwacji.
11
Q
Czym jest efekt grubych ogonów?
A
Prawdopodobieństwa wystąpienia obserwacji znacznie oddalonych od średniej wartości są wyższe niż dla rozkładu normalnego.
12
Q
Czym jest efekt leptokurtozy?
A
Występowanie grubych ogonów i wyższego szczytu funkcji gęstości niż dla rozkładu normalnego.
13
Q
Czym jest efekt dźwigni?
A
Ujemne skorelowanie poziomu kursów i poziomu zmienności stóp zwrotu.
14
Q
Czym są logarytmiczne stopy zwrotu?
A
Definiowane jako ln(yt) - ln(yt-τ).
15
Q
Czy rozkład stopy zwrotu może być rozkładem normalnym?
A
Nie, zazwyczaj nie jest rozkładem normalnym.
16
Q
Czym są efekty grupowania wariancji?
A
Występowanie okresów nasilonej zmienności i okresów stabilnych.
17
Q
Jakie są podstawowe miary zmienności?
A
- Odchylenie standardowe
- Wariancja
- Współczynnik zmienności
18
Q
Czym jest analiza dynamiki zjawisk?
A
Badanie przyrostów absolutnych i względnych w czasie.
19
Q
Czym jest zmienność w kontekście finansowych szeregów czasowych?
A
Mierzy dyspersję lub rozproszenie danych.
20
Q
Czym jest histogram w analizie szeregów czasowych?
A
Pomocny w wstępnej analizie rozkładu.
21
Q
What do the ARCH models describe?
A
Grouping of variance
ARCH models are used in time series analysis to model changing variances over time.
22
Q
What does the leverage effect reflect?
A
Negative correlation between asset prices and volatility of returns
This effect suggests that as asset prices decrease, the volatility of returns tends to increase.
23
Q
What are the models that can be used to test the leverage effect?
A
- EGARCH
- GJR-GARCH
These models incorporate the leverage effect to analyze volatility in financial time series.
24
Q
What is the skewness effect in return distributions?
A
Asymmetry in return distributions, often right-skewed
This effect arises from different investor behaviors during bull and bear markets.
25
How can the skewness effect be modeled?
Using skewed distributions like skewed t-distribution
## Footnote
This helps in accurately representing the asymmetry in the data.
26
What does the autocorrelation effect in returns indicate?
Presence of autocorrelation, especially during low volatility periods
## Footnote
It can be tested using ACF and PACF plots.
27
What models are used to describe observed autocorrelation in return series?
* ARMA
* ARIMA
* ARFIMA
## Footnote
These are standard models in time series analysis for handling autocorrelation.
28
What characterizes the long memory effect in volatility?
Significant high-order autocorrelation coefficients of squared returns
## Footnote
This effect implies that after large increases, further increases are likely followed by sudden decreases.
29
Which model is commonly used to describe the long memory effect?
FIGARCH
## Footnote
The FIGARCH model is specifically designed to capture long memory in volatility.
30
What does the non-trading period effect refer to?
Information accumulation during market closures affecting prices upon reopening
## Footnote
Variance from Friday to Monday can be significantly higher than from Monday to Tuesday.
31
What is the January effect?
An anomaly where stock returns in January are abnormally high
## Footnote
This effect indicates that markets may be inefficient in terms of information.
32
What does the day of the week effect analyze?
Variations in returns, volumes, or price spreads on specific days
## Footnote
This effect is explored to understand trading behaviors throughout the week.
33
What hypothesis is tested when analyzing the day of the week effect?
H0: E(yi) = E(yj)
## Footnote
This null hypothesis tests if the means of returns on different days are equal.
34
What statistical test is used for comparing means in day of the week analysis?
Two-sample mean equality test
## Footnote
This test assesses whether there are significant differences in returns across different days.
35
What does the Fisher-Snedecor statistic test in the context of the day of the week effect?
Whether variances in selected days differ significantly
## Footnote
This involves comparing the sample variances from different days of the week.
36
What are the types of time series in econometric literature?
* Moment series
* Period series
## Footnote
These classifications help in understanding the nature of time series data.
37
What is the definition of forecasting?
Predicting the probable course of events both quantitatively and qualitatively
## Footnote
Forecasting supports decision-making processes.
38
What is a forecast?
The result of the forecasting process
## Footnote
Forecasts can be quantitative (point or interval) or qualitative.
39
What characterizes an econometric forecast?
It is a probabilistic judgment with known likelihood
## Footnote
This reflects the uncertainty inherent in forecasting.
40
What are the functions of forecasting?
* Preparatory
* Informational
* Activating
## Footnote
These functions highlight how forecasts impact decision-making and actions.
41
What are the main stages of forecasting?
* Formulation of the forecasting task
* Determination of forecasting premises
* Data collection and analysis
* Method selection
* Construction of the forecast
* Evaluation of forecast accuracy
## Footnote
These stages ensure a structured approach to forecasting.
42
What are the types of forecasts based on duration?
* Short-term
* Medium-term
* Long-term
## Footnote
Each type has different time horizons depending on the context.
43
What is the structural approach to forecasting?
Uses a structural model reflecting economic theory
## Footnote
It involves specifying analytical forms that represent the underlying economic mechanisms.
44
What is the non-structural approach to forecasting?
Relies on past values of the variable to predict future values
## Footnote
This approach includes naive methods and time series models like ARIMA.
45
What are the principles of quantitative prediction?
* Point prediction
* Unbiased prediction principle
* Maximum likelihood principle
* Minimum expected loss principle
## Footnote
These principles guide the methods used for making predictions.
46
What does the unbiased prediction principle state?
The forecast is set at the expected value of the variable
## Footnote
This principle ensures that the prediction is centered around the true value.
47
What is the principle of maximum likelihood in prediction?
Forecasting aims to maximize the likelihood of observed values
## Footnote
This principle is fundamental in statistical estimation.
48
What does the principle of minimum expected loss entail?
Minimizing the expected loss associated with prediction errors
## Footnote
This principle emphasizes the economic rationale behind forecasting.
49
What is the goal of minimalization of expected loss?
y p = min E(W (u))
## Footnote
Where W(u) is the loss function such that W(0) = 0 and W(u) > 0 when u is not zero (u is the prediction error).
50
What principle is the same in the case of a symmetric distribution of the forecasted variable?
The principle of unbiased prediction and maximum likelihood.
51
What is the requirement for the confidence interval in prediction?
P(y ∈ I p) = γ (γ ≥ 0.90)
## Footnote
Where γT is the credibility of the prediction.
52
What factors do prediction accuracy measures depend on?
* Correct estimation of model parameters
* Application of the appropriate inference principle
* Adoption of the correct initial assumptions.
53
What are the indirect methods of forecasting?
* Descriptive econometric models
* Causal models.
54
What are some direct methods of forecasting?
* Moving averages
* Exponential smoothing
* Development trend models
* Time series models (AR, ARMA, ARIMA, SARIMA, ARFIMA).
55
What is the naive forecasting method according to the constant level?
y p = y t-1.
56
What is the formula for the median in forecasting?
y p = Me(y)
## Footnote
Where Me is the median obtained for an odd number of observations.
57
What is the formula for the modal forecasting method?
y p = Mo(y).
58
How is the average calculated from the minimum and maximum empirical values?
y p = (ymin + ymax) / 2.
59
What is the formula for forecasting using constant absolute increments?
y p = Y + (Y - Y t-1).
60
What is the formula for forecasting using constant relative increments?
y t = y t-1(1 + Δy/y t-1).
61
When can naive methods be used in forecasting?
When there are small random fluctuations (low variability coefficient) and for short-term forecasting.
62
What are the advantages of naive models?
* Simple implementation
* Useful for comparisons with forecasts generated by other methods.
63
What are the disadvantages of naive models?
* Assumes the phenomenon develops at the level of the previous period
* Low dispersion
* Random walk model in the stock market.
64
What is the formula for the moving average model?
y p = (1/k) * Σ(y t-i) for i = 0 to k-1.
65
What is the effect of increasing the smoothing constant in moving averages?
It increases the smoothing effect.
66
What is the forecast formula using a four-period moving average?
F = (Y n-4 + Y n-3 + Y n-2 + Y n-1) / 4.
67
What is the formula for a weighted moving average?
y p = Σ(w i * y t-i) for i = 0 to k-1.
68
What is the limitation of moving averages in forecasting?
Only the k most recent values are considered, excluding earlier observations.
69
What is the formula for the forecast error at time t?
D = y t - y p.
70
What is the formula for the mean error (ME)?
ME = (1/(m-k)) * Σ(y t - y p) for t in m.
71
What does the mean absolute error (MAE) measure?
It informs how much the forecasts deviate from the actual values on average.
72
What is the formula for the mean squared error (MSE)?
MSE = (1/(m-k)) * Σ(y t - y p)² for t in m.
73
What does the root mean square error (RMSE) represent?
It is the standard deviation of forecast errors.
74
What is the formula for the relative forecast error (PE)?
PE = ((y t - y p) / y t) * 100.
75
What does the mean percentage error (MPE) indicate?
It shows the relative bias of the obtained forecasts.
76
What is the formula for the mean absolute percentage error (MAPE)?
MAPE = (1/(m-k)) * Σ(|(y t - y p) / y t|) * 100.
77
What is the range of the Theil coefficient?
[0;1].
78
What components are included in a time series?
* Trend (T)
* Seasonal fluctuations (S)
* Random fluctuations (P)
* Cyclical fluctuations (C).
79
What are the types of relationships between components of a time series?
* Additive model
* Multiplicative model.
80
What should be calculated as part of an empirical example related to stock prices?
* Calculate returns and logarithmic returns
* Compare distributions of stock prices and returns
* Estimate stock price forecasts using at least three naive methods.
81
What is the formula for the naive forecasting method?
yt = yt-1(1 + t)
## Footnote
This formula indicates that the forecast for the current period is based on the previous period's value adjusted by a factor.
82
What is the average fuel consumption (Yt) for the provided data?
10.1
## Footnote
This average is calculated from the fuel consumption values given for weeks 1 to 12.
83
What is the standard deviation of the fuel consumption data?
0.646
## Footnote
This value indicates the dispersion of the fuel consumption data around the mean.
84
What is the coefficient of variation for the fuel consumption data?
6.37
## Footnote
This is a measure of relative variability calculated as the ratio of the standard deviation to the mean.
85
What does RMSE stand for in forecasting?
Root Mean Square Error
## Footnote
RMSE is a measure used to assess the accuracy of a forecasting method.
86
What are the advantages of naive models in forecasting?
* Assumes the phenomenon develops at the level of the previous period
* Low dispersion
* Based on expert opinion
* Frequently used for comparison with other forecasting methods
## Footnote
These advantages highlight the simplicity and ease of use of naive forecasting models.
87
What is the formula for Mean Absolute Error (MAE)?
MAE = (1/(m - k)) * Σ|yt - ypt|
## Footnote
MAE measures the average magnitude of errors in a set of forecasts, without considering their direction.
88
How is the Mean Squared Error (MSE) calculated?
MSE = (1/(m - k)) * Σ(yt - ypt)²
## Footnote
MSE measures the average of the squares of the errors, providing insight into the variance of forecast errors.
89
What does the Theil coefficient indicate?
Indicates the presence of systematic error in forecasting
## Footnote
A Theil coefficient greater than 0.2 suggests a need to re-estimate model parameters.
90
What is the purpose of moving averages in forecasting?
Used for smoothing data and forecasting
## Footnote
Moving averages help to reduce noise and reveal underlying trends in time series data.
91
What is the weighted moving average formula?
yp = Σ(yi * wi)
## Footnote
In this formula, weights are applied to past observations, allowing more recent data to have a greater influence on the forecast.
92
What is the model used for exponential smoothing?
Brown Simple Exponential Smoothing
## Footnote
This model uses previous observations and forecasts to smooth the time series.
93
What is the formula for the exponential smoothing model?
yp = α * yt + (1 - α) * ypt-1
## Footnote
Here, α is the smoothing constant between 0 and 1, determining the weight given to the most recent observation.
94
What is the definition of relative forecast error (PE)?
PE = (yt - ypt) / yt × 100
## Footnote
This formula expresses the error as a percentage of the actual value, providing a relative measure of forecast accuracy.
95
What is the purpose of using a double moving average model?
Used when there is a linear trend in the data
## Footnote
This method applies a moving average to a smoothed series to enhance forecasting accuracy.
96
What does MAPE stand for and how is it calculated?
Mean Absolute Percentage Error
MAPE = (1/(m - k)) * Σ(|yt - ypt| / yt) × 100
## Footnote
It measures the accuracy of a forecasting method as a percentage, allowing easy interpretation.
97
What is a disadvantage of using moving averages for forecasting?
Only considers the last k values of the variable
## Footnote
This limitation can lead to loss of potentially valuable information from earlier observations.
98
What is the significance of weights in weighted moving averages?
Weights determine the influence of each observation on the forecast
## Footnote
Properly assigned weights can significantly enhance forecast accuracy.
99
What is the initial value estimation method referred to in the text?
Average from several initial periods
## Footnote
This method is used to establish a baseline for further calculations.
100
What is the estimation method based on the least squares called?
Estimation using the least squares method (Makridakis and Wheelwright 1989)
## Footnote
This method minimizes the sum of the squares of the residuals.
101
What is the linear exponential smoothing model by Holt used for?
It is used when a time series exhibits a trend and random fluctuations
## Footnote
Holt's model introduces two smoothing parameters: α and β.
102
What are the two equations used in Holt's linear exponential smoothing model?
1. Ft-1 = αyt-1 + (1 + α)(Ft-2 + St-2)
2. St-1 = β(Ft-1 - Ft-2) + (1 - β)St-2
## Footnote
These equations help smooth the time series values and trend increment values.
103
What does the forecast equation for t > n in Holt's model look like?
yp = F + (t - n) × S
## Footnote
This equation predicts future values based on the smoothed values and trend.
104
What are the initial values F1 and S1 for building Holt's model?
F1 = Y1, S1 = Y2 - Y1
## Footnote
These values are essential for initializing the Holt model.
105
What is the Winters model an extension of?
It is a generalization of Holt's method for cases with a seasonal component
## Footnote
The Winters model can be additive or multiplicative depending on seasonal fluctuations.
106
What are the three smoothing constants in the Winters model?
1. α - for level trend smoothing
2. β - for trend change smoothing
3. γ - for seasonal fluctuations smoothing
## Footnote
These constants range within the interval [0,1].
107
What is the forecast equation for the Winters model?
yˆt+h = Tt + hBt + St+h-s
## Footnote
This equation calculates future forecasts considering trend and seasonal components.
108
What are the components of a time series mentioned in the text?
1. Trend (T)
2. Seasonal fluctuations (WS)
3. Cyclical fluctuations (WC)
4. Random fluctuations (WP)
## Footnote
These components can interact in additive or multiplicative ways.
109
What are the two types of dynamic models mentioned?
1. Statistical models
2. Dynamic models
## Footnote
Statistical models examine relationships at a specific time, while dynamic models consider time as a factor.
110
What are the types of dynamic models based on endogenous and exogenous variables?
1. Models with exogenous time variable (t)
2. Models with delayed endogenous variables
## Footnote
These models capture relationships and changes over time.
111
What is the most common form of trend function mentioned?
Linear function: yt = a0 + a1t + εt
## Footnote
This function represents the level of the observed phenomenon over time.
112
What is the purpose of statistical tests such as the Student's t-test mentioned in the text?
To assess the conformity of the trend form with empirical data
## Footnote
These tests help determine the validity of the chosen trend model.
113
What are the methods of extracting a pure trend from time series?
1. Mechanical method (using moving averages)
2. Analytical method (fitting an appropriate function)
## Footnote
These methods aim to eliminate fluctuations over time for clearer trend analysis.
114
What is a key challenge in forecasting using the Winters method?
Determining the values of the model parameters (α, β, γ) and initial values
## Footnote
These challenges are common across adaptive forecasting methods.
115
What is the form of the multiplicative model in the Winters method?
Tt = αyt + (1 - α)(Tt-1 + Bt-1)
## Footnote
This equation adjusts the trend based on seasonal components.
116
What is the purpose of using numerical methods like Newton's or Marquardt's in parameter estimation?
To estimate parameters in nonlinear models
## Footnote
These methods optimize the fitting of the model to the data.
117
What is the form of a power trend model?
yt = a t a1 eεt
## Footnote
(t = 1,..., n)
118
What are the methods for estimating parameters in trend models?
* Methods numeryczne (Newtona, Marquardta, najszybszego spadku)
* MNK, po uprzedniej transformacji zmiennych
119
What transformation is used for the forecast variable in linear trend functions?
y = ln yt
120
What is the matrix representation of the linear trend model?
y = Xa + ε
121
What is the formula for the standard error of estimation?
e = sqrt(1/(n - k - 1) * Σ(ln y - ln yˆ)²)
## Footnote
(t = 1 to n)
122
What does the coefficient of determination (R²) represent?
R² = Σ(ln yˆt - ln y)² / Σ(ln yt - ln y)²
## Footnote
(t = 1 to n)
123
True or False: The coefficient of indeterminacy (φ²) is calculated as φ² = 1 - R².
True
124
What is the general form of a trend model?
yt = f(x1t, x2t, ..., xkt, εt)
## Footnote
where yt is the dependent variable
125
What are the two types of multicollinearity?
* Współliniowość algebraiczna
* Współliniowość statystyczna
126
How is the moving average (SRk) calculated for odd k?
SRk = (y1 + y2 + ... + yk) / k
127
What is the formula for seasonal effects in additive models?
Tt = yt - SRt
128
How do you calculate raw seasonal indices?
S = (1/k) * ΣSj
## Footnote
where k is the number of cycles
129
What is the purpose of seasonal adjustment in time series data?
To remove seasonal components from the original series
130
What is the formula for the forecast in an additive model?
yp = T + St + ht
131
What is the formula for the forecast in a multiplicative model?
yp = T × St × ht
132
What is the significance of the error standard deviation (Se)?
It indicates the accuracy of the estimation in the model.
133
What is the relationship between cyclical fluctuations and seasonal adjustments?
Cyclical fluctuations are determined from a time series adjusted for seasonal variations.
134
What is the transformation for the exponential trend model?
y = a ateεt
## Footnote
(a > 0, a ≠ 1, t = 1,..., n)
135
What does the term 'εt' represent in econometric models?
The stochastic component or error term
