An ARIMA(1, 0, 0) is an AR(1) model. Therefore it is given by

X(t) = aX(t-1) + e(t)

where {e(t)} is a white noise process and a is a constant chosen so that the model might be stationary.

Look up ARDL or ADL models in any elementary econometrics textbook.

Hamideh Kazemi , based on your equation, the value of "b" is the weight of a lagged effect of X on Y. Think for example in the implementation of a public policy (X) on the overall economy (Y), in some situations the effect of X on Y will be contemporaneous (as e.g. a sudden and severe lockdown due to Covid-19), while in other cases the effect will happen with a lag (of some months, for example, like the tax-reduction or income policies to support the activities affected by CV19. You can argue that the effect of income policies based on cash transfers will also have an immediate effect on Y, and hence you have both a contemporaneous effect X(t) on and a lagged effect X(t +h) on Y(t):

Y(t + h) = b*X(t) + a*X(t +h)

In a time -h,

Y(t) = b*X(t-h) + a*X(t),

a is the parameter measuring the contemporaneous effect and b the one measuring the lagged effect. Assuming that you are more interested on the impact in the changes of Y between t = 0 and t = h,

Y(t + h) - Y(t) = b*X(t) + a*X(t +h),

rearranging and solving for -h:

Y(t + h) = Y(t) + b*X(t) + a*X(t +h),

Y(t) = Y(t-h) + a*X(t) + b*X(t-h),

which for h = 1 is your equation.)

Ette Etuk Rolando Gonzales Tihana Škrinjarić

Thanks a lot for your responses.

And may I ask what is the difference between arimax, and "Regression With ARIMA Error"?

Why do we even call it "Arima error" model?

Hamideh Kazemi , and arimaX model is just an arima model with an additional X regressor. For example, an ARIMAX (1,0,0):

y(t) = a + b_1*y(t-1) + b_2*X(t) + e(t)

A regression with arima errors is the one below:

y(t) = a + b_1*X(t) + e(t)

e(t) = b_2*e(t-1) + b_3*e(t-2) + u(t)

for an e(t) error term that follows and ARIMA(2,0,0) process, AR(2) to be precise. Note that in this last model is the error term the one that follows an ARIMA process.

Rolando Gonzales Thanks a lot for the beautiful explanation :)