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I am new to ARIMA, and I am trying to understand these lag plots. Are the following ACF and PACF suggesting that the lag of my time series is 4? If I am wrong, please help me understand these plots.

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    $\begingroup$ The ACF and PACF suggest that there is no significant correlation to be captured by an ARMA model. The autocorrelations are within the 5% significance bands. You could test this with the test statistics in Box.test. You may be interested in this post. $\endgroup$
    – javlacalle
    Commented Jul 23, 2014 at 8:56
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    $\begingroup$ It could be just noise, it might be an autoregressive term at lag 4. See also the links in the first comment here $\endgroup$
    – Glen_b
    Commented Jul 23, 2014 at 11:35
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    $\begingroup$ As mentioned, most likely the series can be characterised as being a white noise process. Identification, applying Box-Jenkins strategy, is a bit of an art, so looking very hard, it could be possible to tentatively entertain a seasonal model, say, a seasonal AR(1) or seasonal MA(1). Don't get me wrong, it's unlikely, but employing an iterative procedure will help you select the most appropriate model. By the way, how many observations of the series are there? Inspecting 9 lags suggests about 36 or so. With less than 50 observations, it's difficult to find structure in the ACF & PACF. $\endgroup$
    – Graeme Walsh
    Commented Aug 5, 2015 at 14:35

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The threshold statistical significance of the autocorrelations has been noted in the comments and in another answer. What looks interesting is that the autocorrelations in Lag 4 and Lag 8 persist also in the Partial ACF.

Reality should come into play at this point: what are these data? By whatever knowledge you have on the process they measure, is it reasonable to expect that the current level should depend on Lags 4 and 8? If yes, the low estimated strength of autocorrelation is not necessarily an artifact, but an indication that said autocorrelation exists but it is low in strength.

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