Time Series Analysis.

Explore 26 formal proofs and analytical renders within the discipline of Time Series Analysis.

Intermediate

Derivation of the Partial Autocorrelation Function (PACF), specifically $ \phi_{22} $, for an AR(2) Model

Exploring the cinematic intuition of Derivation of the Partial Autocorrelation Function (PACF), specifically

\phi_{22}

, for an AR(2) Model.

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Intermediate

Derivation of the Autocorrelation Function (ACF) for an MA(q) Process, demonstrating its Cut-off Property

Exploring the cinematic intuition of Derivation of the Autocorrelation Function (ACF) for an MA(q) Process, demonstrating its Cut-off Property.

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Advanced

Derivation of the Partial Autocorrelation Function (PACF) for an AR(p) Process, demonstrating its Cut-off Property

Exploring the cinematic intuition of Derivation of the Partial Autocorrelation Function (PACF) for an AR(p) Process, demonstrating its Cut-off Property.

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Intermediate

Derivation of the Infinite MA Representation ($Z_t = \sum \psi_j a_{t-j}$) for a Stationary AR(1) Process

Exploring the cinematic intuition of Derivation of the Infinite MA Representation (

Z_t = \sum \psi_j a_{t-j}

) for a Stationary AR(1) Process.

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Advanced

Proof of the Asymptotic Chi-Squared Distribution of the Ljung-Box Test Statistic

Exploring the cinematic intuition of Proof of the Asymptotic Chi-Squared Distribution of the Ljung-Box Test Statistic.

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Advanced

Derivation of the Method of Moments Estimators for an AR(1) Model with a Constant Term

Exploring the cinematic intuition of Derivation of the Method of Moments Estimators for an AR(1) Model with a Constant Term.

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Advanced

Proof that the Minimum Mean Squared Error (MMSE) Forecast is the Conditional Expectation

Exploring the cinematic intuition of Proof that the Minimum Mean Squared Error (MMSE) Forecast is the Conditional Expectation.

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Intermediate

Derivation of the Explicit Formula for the l-step Ahead Forecast $ \hat{Z}_t(l) $ for a Stationary AR(1) Model

Exploring the cinematic intuition of Derivation of the Explicit Formula for the l-step Ahead Forecast

\hat{Z}_t(l)

for a Stationary AR(1) Model.

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Advanced

Derivation of the Variance of the l-step Ahead Forecast Error for an AR(1) Model

Exploring the cinematic intuition of Derivation of the Variance of the l-step Ahead Forecast Error for an AR(1) Model.

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Advanced

Derivation of the Formula for a 95% Prediction Interval for an AR(1) Forecast

Exploring the cinematic intuition of Derivation of the Formula for a 95% Prediction Interval for an AR(1) Forecast.

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Intermediate

Proof that First Differencing Transforms a Random Walk into a Stationary Process

Exploring the cinematic intuition of Proof that First Differencing Transforms a Random Walk into a Stationary Process.

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Advanced

Formal Proof of the General Invertibility Condition for an MA(q) Process (Roots of Characteristic Polynomial)

Exploring the cinematic intuition of Formal Proof of the General Invertibility Condition for an MA(q) Process (Roots of Characteristic Polynomial).

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Advanced

Derivation of the Infinite AR Representation and $\pi$-weights for an Invertible MA(q) Process

Exploring the cinematic intuition of Derivation of the Infinite AR Representation and

\pi

-weights for an Invertible MA(q) Process.

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Advanced

Derivation of the Autocorrelation Function (ACF) for a Mixed ARMA(1,1) Process

Exploring the cinematic intuition of Derivation of the Autocorrelation Function (ACF) for a Mixed ARMA(1,1) Process.

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Advanced

Derivation of the Variance of the l-step Ahead Forecast Error for a General ARMA(p,q) Process

Exploring the cinematic intuition of Derivation of the Variance of the l-step Ahead Forecast Error for a General ARMA(p,q) Process.

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Advanced

Proof that a Logarithmic Transformation Stabilizes Variance when Standard Deviation is Proportional to Mean

Exploring the cinematic intuition of Proof that a Logarithmic Transformation Stabilizes Variance when Standard Deviation is Proportional to Mean.

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Intermediate

Calculation of the First Few $\psi$-weights for a Specific ARMA(1,1) Model

Exploring the cinematic intuition of Calculation of the First Few

\psi

-weights for a Specific ARMA(1,1) Model.

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Advanced

Formal Proof of the General Stationarity Condition for an AR(p) Process (Roots of Characteristic Polynomial)

Exploring the cinematic intuition of Formal Proof of the General Stationarity Condition for an AR(p) Process (Roots of Characteristic Polynomial).

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Time Series Analysis.

Proof that Autocovariance Depends Only on Lag for Weakly Stationary Processes

Derivation of the Autocorrelation Function (ACF) for a White Noise Process

Proof of the Stationarity Condition for an AR(1) Process (|φ| < 1)

Proof of the Invertibility Condition for an MA(1) Process (|θ| < 1)

Derivation of the Mean for a Stationary AR(1) Process

Derivation of the Variance for a Stationary AR(1) Process

Full Derivation of the Yule-Walker Equations for a General AR(p) Process

Solving Yule-Walker Equations to Derive ACF Values for an AR(2) Model

Derivation of the Partial Autocorrelation Function (PACF), specifically \( \phi_{22} \), for an AR(2) Model

Derivation of the Autocorrelation Function (ACF) for an MA(q) Process, demonstrating its Cut-off Property

Derivation of the Partial Autocorrelation Function (PACF) for an AR(p) Process, demonstrating its Cut-off Property

Derivation of the Infinite MA Representation ($Z_t = \sum \psi_j a_{t-j}$) for a Stationary AR(1) Process

Proof of the Asymptotic Chi-Squared Distribution of the Ljung-Box Test Statistic

Derivation of the Method of Moments Estimators for an AR(1) Model with a Constant Term

Proof that the Minimum Mean Squared Error (MMSE) Forecast is the Conditional Expectation

Derivation of the Explicit Formula for the l-step Ahead Forecast \( \hat{Z}_t(l) \) for a Stationary AR(1) Model

Derivation of the Variance of the l-step Ahead Forecast Error for an AR(1) Model

Derivation of the Formula for a 95% Prediction Interval for an AR(1) Forecast

Proof that First Differencing Transforms a Random Walk into a Stationary Process

Formal Proof of the General Invertibility Condition for an MA(q) Process (Roots of Characteristic Polynomial)

Derivation of the Infinite AR Representation and $\pi$-weights for an Invertible MA(q) Process

Derivation of the Autocorrelation Function (ACF) for a Mixed ARMA(1,1) Process

Derivation of the Variance of the l-step Ahead Forecast Error for a General ARMA(p,q) Process

Proof that a Logarithmic Transformation Stabilizes Variance when Standard Deviation is Proportional to Mean

Calculation of the First Few \(\psi\)-weights for a Specific ARMA(1,1) Model

Formal Proof of the General Stationarity Condition for an AR(p) Process (Roots of Characteristic Polynomial)